//Index Terms -- to be used in searches
// NUMBER PROPERTIES
// FACTORING
// FORMULAS
// EXPONENTIAL
// SYSTEMS OF EQUATIONS AND INEQUALITIES
// MATRICES
// SEQUENCES
// SETS
var owl_caption =
'
';
var owl_says =
'';
var small_owl_caption =
'
';
var small_owl_says =
'';
var light_bulb =
'
'
var write_the_definition =
'';
//MISCELLANEOUS NOTATION ETC.
var summation_diagram =
'The parts of the summation symbol are explained in this diagram.
'
+'
';
var summation_properties =
'
Rules for manipulating sums and the summation symbol are:
'
+'
';
//INDEX_SEQUENCES
var sequences_and_function_notation =
'For most functions studied prior to this point in the course, function names have been '
+'letters like f, g, h chosen from the middle of the alphabet and domain elements have '
+'been letters like x, y, and z chosen from near the end of the alphabet.
'
+'The names of generic sequences usually have names like a, b, c, and d chosen from the '
+'beginning of the alphabet and domain elements are usually denoted by n to correspond with '
+'and emphasize the fact that the domain of a sequence is the set of natural numbers. '
+'However, many special sequences have different names which have become standard '
+'for those sequences.
'
+'Historical usage also suggests a slight variation in function notation as shown in the '
+'following two diagrams.
'
+'
'
+'
'
+'The variation is to write the domain element as a subscript on the '
+'function name rather than to enclose it in parenthesis.';
var terms_of_sequence =
'When working with sequences, range elements are frequently called terms of the sequence.
'
+'For example:
'
+' '
+' - The range element associated with the domain element 1 is called the first term of the sequence.
'
+' - The range element associated with the domain element 6 is called the sixth term of the sequence.
'
+' - The range element associated with the domain element n is called the nth term of the sequence.
'
+'
'
+' '
+' - Because the domain of every sequence is N, the graph of a sequence will consist of a set of discrete points.
'
+' - Because the domain of every sequence is N, it is possible to speak of the first domain element.
'
+' - Because the domain of every sequence is N, it is possible to speak of the next domain element.
'
+' - Because the domain of every sequence is N, it is not possible to speak of the last domain element.
'
+' - Because the domain of every sequence is N, it is possible to ask about and compute the sum of the first k terms. This is usually called the kth partial sum of the sequence.
'
+'
'
+'
';
//INDEX_FORMULAS
var formula_meaning =
'In mathematics a formula (plural: formulae, formulæ or formulas) is a concise way of symbolically '
+'expressing a general relationship between quantities. In beginning algebra courses that relationship is always '
+'expressed as an equation.
Thus for the purpose of this course, a formula must be an equation.';
var formula_square_area =
'The formula for the area of a '
+' '
+' with length x is A = x2.';
var formula_parallelogram_area =
'The formula for the area of a '
+' '
+' with base x and height y is A = xy.';
var formula_rectangle_area =
'The formula for the area of a '
+' '
+' with length x and width y is A = xy.';
var formula_rectangle_perimeter =
'The formula for the perimeter of a '
+' '
+' with length x and width y is P = 2x + 2y.';
var formula_rectangular_solid_volume =
'The formula for the volume of a rectangular solid of length a, width b, and height c is given by V = abc.';
var formula_rectangular_solid_surface_area =
'The formula for the surface area of a rectangular solid of length a, width b, and height c is given by
'
+'Surface Area = 2(ab) + 2(ac) + 2(bc).';
var formula_triangle_area =
'The formula for the area of a '
+' '
+' with base b and height h is 
.';
var formula_trapezoid_area =
'The formula for the area of a '
+' '
+' with bases B and b and height h is
.';
var formula_circle_area =
'The formula for the area of a '
+' '
+' with '
+' '
+'r is: 
.';
var formula_circle_circumference =
'The formula for the circumference of a '
+' '
+' with '
+' '
+'r is: 
.';
var formula_sphere_surface_area =
'The formula for the surface area of a sphere with radius r is '
+'
.';
var formula_sphere_volume =
'The formula for the volume of a sphere with radius r is '
+'
.';
var formula_cylinder_volume =
'The formula for the volume of a cylinder with radius r and height h is '
+'
.';
var formula_cone_volume =
'The formula for the volume of a cone with radius r and height h is '
+'
.';
var formula_temperature_f_to_c =
'To convert temperature measured in degrees Fahrenheit to degrees Celsius, use the formula:'
+'
';
var formula_temperature_c_to_f =
'To convert temperature measured in degrees Celsius to degrees Fahrenheit, use the formula:'
+'
.';
var formula_temperature_c_to_k =
'To convert temperature measured in degrees Celsius to degrees Kelvin, use the formula:'
+'
.';
var formula_distance_between_points =
'The distance d between two points (x1, y1) and (x2, y2) '
+'is given by the formula:
'
+'
.';
var formula_midpoint_between_points =
'The midpoint of the line segment joining two points (x1, y1) and (x2, y2) '
+'is given by the formula:
'
+'
.';
var formula_pythagorean_theorem =
'If a and b are the lengths of the legs of a right triangle with hypotenuse of length c, then
'
+'a2 + b2 = c2.';
var formula_percent =
'Percentage = (Percent)(Base).
If we call Percentage the Amount, the formula becomes '
+' Amount = (Percent)(Base) and if we then replace the words with single letters we obtain a very usable '
+'formula which relates percentage, percent, and base.
'
+'A = PB.
which is read as
'
+' The percentage A is P percent of the base B.';
var percent_problems_three_kinds =
'There are only three percent related problems:'
+''
+'- Given Percent and Base, calculate Percentage
'
+'- Given Percent and Percentage, calculate Base
'
+'- Given Percentage and Base, calculate Percent
'
+'
';
var formula_distance_traveled =
'The distance d traveled at a rate r in time t is given by d = rt.';
var formula_simple_interest =
'The interest I earned by a principle P at a rate r for a time t is I = Prt.';
//END OF FORMULAS
var less_than_facts=
'Another more precise and more algebraic definition of less than is that a is less than b if b - a is positive.
'
+'The expression a < b means a is less than b.
'
+'The expression a < b means a is to the left of b on the number line.
'
+'The expression a < b means b - a is positive.';
var greater_than_facts=
'Another more precise and more algebraic definition of greater than is that a is greater than b if a - b is positive.
'
+'The expression a > b means a is greater than b.
'
+'The expression a > b means a is to the right of b on the number line.
'
+'The expression a > b means a - b is positive.'
// SETS
var set_element_circular=
' Notice that the above two definitions are circular in nature. That is, each uses the other. Therefore, we have no absolute definition of set nor do we have an absolute definition of element of a set. Every logical discussion must have at least one undefined term, and in our discussion we will leave set and element of a set as two undefined terms. The above two definitions will be sufficient for an understanding of everything we will do with sets.';
var set_not_numbers =
'Avoid thinking of sets as collections of numbers. The elements of a set may be things like: people, books, numbers, matricies, functions, equations, rational numbers, desks, polynomials, polynomial equations, polynomial functions, candy bars, clouds, cars, buses, trains, states in the US, countries of the world, computers, geometric figures, essays, test scores, jokes, minerals, trees, flowers, etc.';
var set_mathematical_creatures =
'In this course most of the sets we will discuss will be sets of mathematical creatures (objects) but not necessarily numbers. There are very many mathematical creatures other than numbers.';
var set_notation =
'It is common practice to use symbols to make mathematical discussions more precise and more concise. Some of the symbols, notation and terminology used to discuss sets pervades all of mathematics.
'
+'It is conventional to use capital letters to name sets. Some of these letters are reserved for special sets while others are defined only for the duration of a particular discussion. For example R is reserved for the set of Real Numbers and N is reserved for the set of Natural Numbers. These are conventions only, so the reservation is not absolute. For example Q is reserved for the set of Rational Numbers but Q is also used for other sets when there is no chance for confusion.
'
+'It is conventional to use lower case letters to name elements of a set. This is a convention only and may be violated at times.';
var incorrect_use_of_set_symbols=
'If k is an element of a set K, it is meaningless to write k 
K.
'
+'If k is an element of a set K, it is correct to write k 
K.
'
+'If A is a subset of B, it is meaningless to write A B.
'
+'If A is a subset of B, it is correct to write A B.';
var empty_subset=
'The empty set 
'
+'is a subset of every set. Can you explain why this is true?
Note this implies that every set has at least one subset. You should wonder: '
+'How many subsets does a given set have?';
var subset_of_self=
'Every set is a subset of itself. Can you explain why this is true?
'
+'Note we now know that every set, except the empty set, has at least two subsets. '
+'Why is there an exception for the empty set?';
//END OF SETS
//INDEX NUMBER PROPERTIES
var one_primality =
'The natural number 1 is neither a prime number nor a composite number.';
var rational_decimal =
'The decimal representation of a rational number is a terminating or a repeating decimal.';
var irrational_example =
'Numbers like pi, square root of 2, and the cube root of 5 are examples of irrational numbers.';
var irrational_not_rational =
'A real number which is not rational is irrational.';
var rational_not_irrational =
'A real number which is not irrational is rational.';
var irrational_decimal =
'The decimal representation of an irrational number is neither terminating nor repeating.';
var subset_chain_real =
'The following are important subset relationships between the various subsets of the Real Numbers.
'
+'
'
+'Furthermore 
'
+' and 
'
+' and 
'
+' and 
'
+'
';
var subset_chain_complex =
'The following are important subset relationships between the various subsets of the Complex Numbers.
'
+'
'
+'Furthermore '
+' and '
+' and '
+' and '
+'
';
var property_inheritance =
'When a statement (property, rule, axiom, theorem) is made about the numbers in a particular set of numbers'
+' that property is true for all subsets of numbers contained in the set.';
var special_subsets =
'Some special types of subsets of the Real Numbers:
'
+'
'
+'
';
var subsets_of_numbers_venn1 =
'A chart showing the relationship between various sets of Complex Numbers .
'
+' If two sets are joined by lines the lower set is a subset of the uppermost set.
'
+'
';
var sign_of_product =
'The product of two real numbers with the same signs is positive.
'
+' The product of two real numbers with different '
+'signs is negative.';
var sign_of_quotient =
'The quotient of two real numbers with the same signs is positive.
'
+' The quotient of two real numbers with different '
+'signs is negative.';
var division_of_zero =
'The quotient of 0 divided by any non-zero real number is 0.
';
var division_by_zero =
'Division by zero is undefined.
';
var order_of_operations =
'If no grouping symbols are involved algebraic expressions MUST be evaluated using the following order:'
+''
+'- Evaluate all exponential expressions
'
+'- Do all multiplications and divisions as they occur from left to right
'
+'- Do all additions and subtractions as they occur from left to right
'
+'
'
+''
+'If grouping symbols are involved, remove the grouping symbols by evaluating the expressions inside each pair of grouping symbols '
+'using the following order:'
+'
'
+'- Evaluate all exponential expressions '
+'
'
+'- Do all multiplications and divisions as they occur from left to right'
+'
'
+'- Do all additions and subtractions as they occur from left to right
'
+'
'
+'When grouping symbols are nested, remove the innermost grouping symbols first -- '
+'work from the inside to the outside.';
var transitive_property_of_inequality =
'If a, b, and c are real numbers such that a < b and b < c, then a < c.';
var additive_property_of_inequality =
'If a, b, c, and d are real numbers such that a < b and c < d, then a + c < b + d.';
var multiplicative_property_of_inequality =
'If a and b are real numbers such that a < b and c is a positive real number, then ac < bc
'
+'If a and b are real numbers such that a < b and c is a negative real number, then ac > bc.';
var reflexive_property_of_equality =
'If a is a real number, then a = a.';
var symmetric_property_of_equality =
'If a and b are real numbers and a = b, then b = a.';
var transitive_property_of_equality =
'If a, b, and c are real numbers such that a = b and b = c, then a = c.';
var additive_property_of_equality =
'If a, b, and c are real numbers and a = b , then a + c = b + c.';
var multiplicative_property_of_equality =
'If a, b, and c are real numbers and a = b , then ac = bc.';
var distributive_property =
'If a, b, and c are real numbers, then a(b + c) = ab + ac.';
var commutative_property_of_addition =
'If a and b are real numbers, then a + b = b + a.';
var commutative_property_of_multiplication =
'If a and b are real numbers, then ab = ba.';
var associative_property_of_addition =
'If a, b, and c are real numbers, then (a + b) + c = b + (a + c).';
var additive_identity =
'If a is any real number, then a + 0 = 0 + a = a. The number 0 is called the additive identity.';
var multiplicative_identity =
'If a is any real number, then a(1) = (1)a = a. The number 1 is called the multiplicative identity.';
var additive_inverse_existence =
'If a is any real number then its additive inverse -a exists and a + (-a) = (-a) + a = 0.'
+'
The additive inverse of a number is also called the opposite of the number.';
var multiplicative_inverse_existence =
'If a is any real number, other than 0, then its multiplicative inverse 1/a exists and a (1/a) = (1/a)a = 1.'
+'
The multiplicative inverse of a number is also called the reciprocal of the number.';
var associative_property_of_multiplication =
'If a, b, and c are real numbers, then (ab)c = b(ac).';
var law_of_trichotomy =
'If a and b are real numbers, then one and only one of the following is true:
'
+'
'
+'- a < b
'
+'- a = b
'
+'- a > b
'
+'
';
var trichotomy_and_absolute_value =
'As a result of the Law of Trichotomy there are exactly three possibilities when comparing the absolute value '
+' of an expression with a number.'
+''
+'- |expression| < k
'
+'- |expression| = k
'
+'- |expression| > k
'
+'
';
var trichotomy_and_quadratic_functions =
'Consider a quadratic function whose rule is written in the general form'
+' f(x) = ax2 + bx + c.
From the Law of Trichotomy '
+'it follows that exactly one of the following is true about the leading coefficient:'
+''
+'- a < 0 and the parabola opens down.
'
+'- a = 0 and the function is not quadratic.
'
+'- a > 0 and the parabola opens up.
'
+'
'
+'Also from Law of Trichotomy '
+'it follows that exactly one of the following statements about the discriminant is true:'
+''
+'- b2 - 4ac < 0 and the parabola has no x-intercepts.
'
+'- b2 - 4ac = 0 and the parabola has exactly one x-intercept.
'
+'- b2 - 4ac > 0 and the parabola has two x-intercepts.
'
+'
'
+'Pairing each of the two possibilities for the leading coefficient '
+'with each of the three possibilities for the discriminant, we find the following six '
+' graphical configurations to be the only possibilities for the graph of a quadratic function.
'
+'
'
+'
'
+'
'
+'
'
+'
'
+'
';
var trichotomy_and_quadratic_equation =
'Consider a quadratic equation'
+' y = ax2 + bx + c.
From the Law of Trichotomy '
+'it follows that exactly one of the following is true about the leading coefficient:'
+''
+'- a < 0 and the parabola opens down.
'
+'- a = 0 and the function is not quadratic.
'
+'- a > 0 and the parabola opens up.
'
+'
'
+'Also from Law of Trichotomy '
+'it follows that exactly one of the following statements about the discriminant is true:'
+''
+'- b2 - 4ac < 0 and the parabola has no x-intercepts.
'
+'- b2 - 4ac = 0 and the parabola has exactly one x-intercept.
'
+'- b2 - 4ac > 0 and the parabola has two x-intercepts.
'
+'
'
+'Pairing each of the two possibilities for the leading coefficient '
+'with each of the three possibilities for the discriminant, we find the following six '
+' graphical configurations to be the only possibilities for the graph of a quadratic equation in two variables.
'
+' '
+' '
+' '
+' '
+' '
+'';
var square_root_of_perfect_square =
'The principal square root of a perfect square is a natural number.';
var square_root_of_non_perfect_square =
'The principal square root of a natural number which is not a perfect square is an irrational number.';
//END OF NUMBER PROPERTIES
//INDEX_EXPONENTIAL
var the_number_e =
'An important irrational number is named e. The first few digits in a '
+ 'decimal expression of e are given here: e ≈2.71828184590452
'
+'Other irrational numbers encountered earlier have had understandable explanations. '
+'It is easy to understand what is meant by the square root of 2. It may not be clear why '
+'it is irrational, but at least its meaning is clear. In the same way the number '
+'
is '
+'easily understood to be the ratio of the circumference of a circle divided by its '
+'diameter. The number e on the other hand has no simple explanation.';
var exponential_function_fact1 =
'For each exponential function, the y-intercept is 1.';
var exponential_function_fact2 =
'For each exponential function, there are no x-intercepts and thus no real zeros.';
var exponential_function_fact3 =
'For each exponential function, the graph is entirely above the x-axis. That means '
+'expa(x)>0 for all real numbers x.';
var exponential_function_fact4 =
'For each exponential function, no horizontal line intersects the graph in more than one point.'
+'That means each expa passes the horizontal line test and therefore each exponential function '
+'has an inverse.';
var exponential_function_fact5 =
'For each exponential function, the x-axis is a horizontal asymptote.';
var logarithm_function_fact1 =
'For each logarithm function, there is no y-intercept.';
var logarithm_function_fact2 =
'For each logarithm function, the x-intercept is 1. So each logarithm function has a real zero of 1. ';
var logarithm_function_fact3 =
'For each logarithm function, the graph is entirely to the right of the y-axis. That means '
+'the domain of each logarithm function is the positive real numbers.';
var logarithm_function_fact4 =
'For each logarithm function, no horizontal line intersects the graph in more than one point.'
+'That means each logarithm function passes the horizontal line test and therefore each logarithm function '
+'has an inverse, namely the corresponding exponential function.';
var logarithm_function_fact5 =
'For each logarithm function, the y-axis is a vertical asymptote.';
var identification_of_real_and_complex_components=
'The complex component of a complex number written in standard form is the coefficient'
+'of the imaginary unit i.
The real component of a complex number written in'
+' standard form is the real number which is not the coefficient of i.';
var fact_exponential_meaning =
'When n is a natural number, the exponential expression an is used to '
+'indicate that the base a is multiplied times itself n times.';
var fact_negative_base_even_exponent =
'A negative number with an even exponent is a positive number.';
var fact_negative_base_odd_exponent =
'A negative number with an odd exponent is a negative number.';
var fact_negative_sign_in_base =
'If the base of an exponential includes a minus sign, such as -5 or -x, then that base MUST be enclosed '
+'in parenthesis.
For example, if -5 is to be raised to the third power it must be written as (-5)3';
var exponent_product_of =
'If m, n and a are real numbers then '
+'aman = am + n.';
var exponent_quotient_of =
'If m, n and a are real numbers then '
+'
.';
var zero_exponent =
'If a is not zero then '
+'a0 = 1.';
var negative_one_exponent =
'If a is not zero then '
+'
.';
var negative_exponent =
'If a is not zero and n is a real number then '
+'
.';
var exponent_of_product =
'If a, b and m are real numbers then '
+'(ab)m = ambm.';
var exponent_of_quotient =
'If a, b and m are real numbers then '
+'
.';
var exponent_of_exponential =
'If a, m and n are real numbers then '
+'(am)n = amn.';
var fact_fraction_three_signs_keyword =
'
'
+'Three Signs of a Fraction';
var fact_fraction_three_signs_caption =
'Definition
Three Signs of a Fraction';
var fact_fraction_three_signs =
'The sign of the fraction, the sign of the numerator, and the sign of the denominator produce four equivalent forms '
+'of every fraction.
'
+'
';
var one_as_fraction =
'If c represents a non-zero real number then '
+'
';
var improper_fraction_to_mixed_number =
'To convert an improper fraction to a mixed number, divide the numerator by the denominator. '
+'The quotient is the whole number part of the mixed number and the remainder is the numerator of the '
+'fractional part of the mixed number.';
var mixed_number_to_improper_fraction =
'To convert a mixed number to an improper fraction, write the whole number as a '
+'fraction and perform the indicated addition.';
var mixed_number_arithmetic =
'If a mixed number is involved in any addition, subtraction, multiplication or division, change that mixed number '
+'to an improper fraction before proceeding.';
var fundamental_property_of_fractions =
'If a represents a real number and b and c represent non-zero real numbers then '
+'
';
var fundamental_property_of_fractions_fact =
'The Fundamental Property of Fractions is as valid for rational expressions as it is for common fractions. The only '
+ 'difference is that when applying the property to rational expressions one must interpret a, b and c as polynomials.';
var product_of_fractions =
'If a, b, c, and d represent real numbers and b is not 0 and d is not 0 then '
+' 
';
var product_of_fractions_fact =
'The Rule for Multiplying Fractions is as valid for rational expressions as it is for common fractions. The only '
+'difference is that when applying the rule to rational expressions one must interpret a, b,c and d as polynomials.';
var product_of_fractions_remembering =
'A good way to remember this rule is to observe that it states:
'
+' The numerator of a product is the product of the numerators and
'
+' The denominator of a product is the product of the denominators.';
var product_of_fractions_alert =
'There is no requirement to find and/or use common denominators when multiplying fractions.';
var quotient_of_fractions =
'If a represents a real number and b, c, and d represent non-zero real numbers, then '
+'
';
var quotient_of_fractions_fact =
'The definition of division as applied to fractions is as valid for rational expressions as it is for '
+'common fractions. The only difference is that when applying the rule to rational expressions one must '
+'interpret a, b,c and d as polynomials.';
var quotient_of_fractions_diagram =
'An alternate depiction of this definition of division whether applied to fractions or rational expressions is '
+'this familiar diagram
'
+'
';
var sum_of_fractions =
'If a, b, and d represent real numbers and d is not 0 then '
+' 
';
var sum_of_fractions_two =
'If a, b, c, and d represent real numbers and neither b nor d is 0 then '
+'the sum 
is computed by '
+'reducing or expanding the two fractions to fractions with the same denominator and then adding those two '
+'fractions with the same denominator as described above.';
var sum_of_fractions_two_fact =
'Adding fractions with the different denominators is as valid for rational expressions as it is for common fractions. '
+'The only difference is that when applying the rule to rational expressions one must interpret a, b, c, and d as '
+'polynomials. ';
var difference_of_fractions =
'Subtraction of fractions is defined as an addition as shown here. '
+'
';
var difference_of_fractions_process =
'To subtract one fraction or rational expression from another, change the problem to an addition problem and '
+'proceed according to the rules for addition.';
var expanding_fraction =
'To expand a fraction choose a fraction with equal numerator and denominator such '
+'that the product of its denominator and the denominator we are expanding is the desired new '
+'denominator. The product of this newly created fraction and the original fraction is the '
+'desired expanded fraction.';
var sum_of_fractions_fact =
'The definition of addition as applied to fractions is as valid for rational expressions as it is for '
+'common fractions. The only difference is that when applying the rule to rational expressions one must '
+'interpret a, b,and d as polynomials.';
var sum_of_fractions_remembering =
'A good way to remember this rule is to observe that it states:
'
+' The numerator of a sum is the sum of the numerators and
'
+' The denominator of a sum is the common denominator.';
var simplify_complex_fraction =
'To simplify a complex fraction simply perform the indicated division.';
var simplify_complex_fraction_with_sums =
'If any of the numerators or denominators of the fractions contain a sum or difference, the indicated addition '
+'(or subtraction) should be performed before attempting the division.';
var simplify_complex_fractions_fact =
'Simplifying complex fractions is as valid for rational expressions as it is for '
+'common fractions. The only difference is that when applying the rule to rational expressions one must '
+'interpret numerators and denominators as polynomials.';
var simplify_complex_fractions_alternate =
'An alternate process for simplifying rational expressions is to multiply both the numerator and denominator '
+'by the LCD of the rational expressions contained in the numerator AND the denominator. ';
var adding_like_terms =
'To add like terms, add their coefficients and keep the same variables with the same exponents.
'
+ 'Adding like terms is frequently called combining like terms.
'
+'Adding like terms is a direct application of the Distributive Property.';
var equation_circle =
'The equation of a circle with radius r and center at the origin is
x2 + y2 = r2
'
+'while the equation of a circle with radius r and center at the point (h, k) is
(x - h)2 + (y - k)2= r2.
'
+'The latter equation is called the Standard Form for the equation of a circle.
'
+'If the Standard Form for the equation of a circle is expanded, '
+'The General Form of the equation of a circle
x2 + y2 + Dx + Ey + F = 0
'
+'is obtained.';
var general_quadratic_two_variables_circle =
'The General Quadratic Equation in two variables
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
'
+'is the equation of a circle if and only if A = C and B = 0.';
var general_form_to_standard_form =
'The following process may be used to convert the General Form to the Standard Form so that the center and radius of the circle are obvious.
'
+''
+'- Write an equivalent equation with variable terms on the left side and constants on the right side.
'
+'- Complete the square on the x-terms and create an equation equivalent to the original by adding the same constant to the right side of the equation.
'
+'- Complete the square on the y-terms and create an equation equivalent to the original by adding the same constant to the right side of the equation.
'
+'- Factor both perfect square trinomials to arrive at the Standard Form.
'
+'
'
+' The center and radius are now easily observed.';
var pick_circle_origin=
'Since the center is the origin (0,0) we use the equation
x2 + y2 = r2.';
var find_radius_pick_circle_general=
'Since the center is not the origin we use the equation
(x - h)2 + (y - k)2= r2. '
+'However, before we can use that equation we must find the radius. The radius is the distance from the center to '
+' a point on the circle. We find the radius by using the distance formula to calculate the distance between
(3, -2) and (-1, 1).
'
+' Recall the distance formula is: 
' ;
var pick_circle_general=
'Since the center is not the origin we use the equation
(x - h)2 + (y - k)2= r2.';
var testing_a_point =
'A point satisfies an equation if a true statement results when the coordinates of the point are substituted into the equation.
'
+'If a true statement results, the point is on the graph of the equation.
'
+'If a false statement results, the point is not on the graph of the equation.';
var testing_a_solution =
'Substitute the number into the equation.
'
+'If a true statement results, the number is a solution of the equation.
'
+'If a false statement results, the number is not a solution of the equation.';
var finding_x_intercept =
'The x-intecept of the graph of an equation is found by setting y = 0 in the equation and solving for x.
'
+'This works because every point on the x-axis has 0 as its y-coordinate.';
var finding_y_intercept =
'The y-intecept of the graph of an equation is found by setting x = 0 in the equation and solving for y.
'
+'This works because every point on the y-axis has 0 as its x-coordinate.';
var generating_equivalent_equations1 =
'If any expression is added to both sides of an equation the resulting equation '
+'is equivalent to the original equation.';
var generating_equivalent_equations2 =
'If both sides of an equation are multiplied by the same non-zero real number, the resulting equation '
+'is equivalent to the original equation.';
var generating_equivalent_equations1and2 =
'Important Property 1:If any expression is added to both sides of an equation the resulting equation '
+'is equivalent to the original equation.
'
+'Important Property 2:If both sides of an equation are multiplied by the same non-zero real number, the resulting equation '
+'is equivalent to the original equation.';
var linear_equation_one_variable_solving =
'The process to solve a linear equation in one variable is to generate a sequence of equations each '
+'equivalent to the previous equation until a simplest equation is obtained.';
var number_of_solutions_linear_equation_one_variable =
'A linear equation in one variable will have no solutions, one solution, or have all real numbers as solutions.'
var comment_about_solving_linear_equations =
'Start with the original equation (the one to be solved) and use the above two properties of equations to generate simpler '
+'equations, all equivalent to the original equation, until a simplest equation is obtained.
'
+'Recall that equivalent equations have the same solution set. Therefore the simplest equation found by the '
+'above process has the same solution set as the original equation. The solution set for the simplest equation '
+'is clearly the single number on one side of the equation.
'
+'If the equation is not well formed, it is advisable to simplify the expression on each side of the equality '
+'symbol before beginning to use the above two properties. Adhere to the order of operations when simplifying '
+'each side.';
var linear_equation_one_variable_graph =
'The '
+' '
+' for conditional linear equation in one variable is a set containing a single number. '
+'The '
+' '
+' of a conditional linear equation in one variable is a single point on the Real Number Line.'
+'The solution set for a linear '
+' '
+' in one variable is the empty set. A linear contradiction in one variable has no graph.
'
+'The solution set for a linear '
+' '
+' in one variable is all Real Numbers. The graph of '
+'a linear identity in one variable is the entire Real Number Line.
';
var linear_inequality_one_variable_graph =
'The graph of a linear inequality in one variable is the empty set, a ray, or the Real '
+'Number Line.';
var linear_inequality_two_variable_graph =
'';
var square_root_of_both_sides =
'If one takes the square root of both sides of an equation the resulting equations need not be '
+'equivalent to the original equation. However, the solution set of the resulting equation contains the '
+'solution set of the original equation but it may also contain numbers which are not solutions of the'
+' original equation.
An important part of the solving process in this case involves substituing '
+'the possible solutions into the original equation to determine which numbers are actually solutions.';
var multiply_both_sides_by_variable1 =
'When both sides of an equation are multiplied by an expression which contains a varible, the resulting equation '
+'is generally NOT equivalent to the original equation.';
var multiply_both_sides_by_variable2 =
'When both sides of an equation are multiplied by an expression which contains a varible, the solution set of the '
+'resulting equation CONTAINS the solution set of the original equation.';
var domain_of_rational_expression =
'The domain of a rational expression, unless otherwise stipulated, '
+'is the set of all real numbers for which the denominator is not zero.';
var solve_a_rational_equation =
'To solve a rational equation;
'
+'1) Multiply both sides of the equation by the LCD of all rational expressions in the equation.'
+'
'
+'2) Solve the resulting equation to obtain possible solutions of the original equation.'
+'
'
+'3) Determine the solution set of the original equation by checking each of the possible '
+'solutions in the original equation.' ;
var solve_a_rational_equation_fact1 =
'If both sides of a rational equation are multiplied by the LCD of the rational expressions in the equation '
+'and if that LCD is a number (no variables), then the resulting equation is equivalent to the original equation.';
var solve_a_rational_equation_fact2 =
'The only way a possible rational solution (as found by the above process) can fail to be a solution of the original '
+'equation is for the possible solution to cause a denominator in the original equation to be zero . Consequently, '
+'possible solutions can be checked by simply insuring that they do not cause any denominator to be zero. ';
var solve_a_rational_equation2 =
'To solve a rational equation;
'
+'1) Add or subtract appropriate rational expressions to both sides of the equation to obtain 0 on one side of the equation.'
+'
'
+'2) Perform all the indicated additions and subtractions of rational expressions.'
+'
'
+'3) Use the Zero Factor Property to conclude the numerator must be zero.'
+'
'
+'4) Solve the resulting equation to obtain possible solutions of the original equation.'
+'
'
+'5) Determine the solution set of the original equation by checking each of the possible '
+'solutions in the original equation.' ;
var compound_inequality_solution_intersection =
'The solution set of a compound inequality formed with the word and is the intersection of '
+'the solution sets of the two inequalities.';
var compound_inequality_solution_union =
'The solution set of a compound inequality formed with the word or is the union of '
+'the solution sets of the two inequalities.';
var compound_inequality_compact_notation =
'When using the compact form of a compound inequality such as a < x < b, '
+'be careful to write only meaningful statements. The inequalities must both "point" '
+'in the same direction. The end expressions must be related as indicated by the inequality symbols. '
+'For example, in a < x < b, it must be true that a '
+'< b.';
var compound_inequality_absolute_value1 =
' If k is a positive number and X is either a single variable or a variable expression, then the inequality
'
+' '
+'| X | < k is equivalent '
+'to - k < X < k
';
var compound_inequality_absolute_value2 =
' If k is a positive number and X is either a single variable or a variable expression, then the inequality
'
+' '
+' | X | > k is equivalent to X < - k OR X > k
'
+'Note this is a compound inequality formed with the word OR.
'
+'Therefore the solution set is the union of the solution sets of the two individual inequalities.';
var generating_equivalent_inequalities1 =
'';
var generating_equivalent_inequalities2 =
'';
var generating_equivalent_inequalities3 =
'';
var generating_equivalent_inequalities1and2and3=
'Important Property 1:If any expression is added to both sides of an inequality the resulting inequality '
+'is equivalent to the original inequality.
'
+'Important Property 2:If both sides of an inequality are multiplied by the same positive real number, the resulting inequality '
+'is equivalent to the original inequality.
'
+'Important Property 3:If both sides of an inequality are multiplied by the same negative real number and the inequality symbol is reversed, '
+'the resulting inequality is equivalent to the original inequality.';
var solving_inequality_in_one_variable =
'Start with the original inequality (the one to be solved) and use the above three properties to generate '
+'simpler inequalities, all equivalent to the original inequality, until a simplest inequality is obtained. ';
var linear_inequality_graph =
'Interval notation is used to express and graph the solution set of an inequality. The various possibilities for intervals are shown in the table below'
+'
'
+'
';
var linear_inequality_graph2 =
'Interval notation is used to express and graph the solution set of an inequality.'
+'The various possibilities for intervals are shown in the table below'
+'';
var quadratic_equation_graph_of =
'The graph of a quadratic equation in two variables is a parabola which opens up if '
+'a > 0 and opens down if a < 0.';
var quadratic_equation_intercepts_of =
'The y-intercept of the graph of a quadratic equation in two variables '
+'y = ax2 + bx + c is (0, c).
'
+'The x-intercepts of the graph of a quadratic equation in two variables '
+'y = ax2 + bx + c are found by solving the corresponding '
+'quadratic equation, 0 = ax2 + bx + c, in one variable.';
var quadratic_equation_vertex_of =
'The first coordinate of the vertex of the graph of a quadratic equation in two '
+'variables y = ax2 + bx + c
is -b/2a.';
var quadratic_equation_discriminant_of =
'The discriminant of a quadratic equation in two variables y = ax2 + bx + c '
+' is b2 - 4ac.
'
+'If the discriminant is positive,the graph has two x-intercepts.
'
+'If the discriminant is zero,the graph has one x-intercept.
'
+'If the discriminant is negative,the graph has no x-intercepts and in this case,
'
+' The graph is entirely above the x-axis if a > 0.
'
+' The graph is entirely below the x-axis if a < 0.';
var quadratic_equation_one_variable_discriminant_of =
'The discriminant of a quadratic equation in one variable'
+' ax2 + bx + c = 0 '
+' is b2 - 4ac.
'
+'If the discriminant is positive,the equation has two real solutions.
'
+'If the discriminant is zero,the equation has one real solution.'
+'
If the discriminant is negative,the graph has no '
+'real solutions and in this case,
'
+'the equation has two complex solutions. Furthermore the complex solutions are complex conjugates. ';
var discriminant_and_law_of_trichotomy=
'Observe that in the above fact, the discriminant is compared to zero.'
+' According to the Law of Trichotomy exactly one of the following is true:'
+''
+'- the discriminant is greater than 0 (is positive),
'
+'- the discriminant is equal to 0,
'
+'- the discrimanant is less than zero (is negative).
'
+'
'
+'Therefore according to The Law of Trichotomy all possibilities are considered in the above'
+' fact.';
var quadratic_formula =
'The solutions of a quadratic equation ax2 + bx + c = 0 are given by
'
+'
';
var quadratic_solve_by_factoring =
'Many quadratic equations in one variable may be solved by using factoring techniques in conjunction with'
+' the Zero Factor Property as described here:
'
+''
+'- Write the quadratic equation in standard form.
'
+'- Factor the quadratic polynomial into a product of linear factors.
'
+'- Use the Zero Factor Property to set each factor equal to zero.
'
+'- Solve each of the resulting linear equations.
'
+'
'
+'The resulting solutions are solutions of the original quadratic equation.';
var equation_solve_by_factoring =
'
Some equations may be solved by using factoring techniques in conjunction with'
+' the Zero Factor Property as described here:'
+''
+'- Write the equation in standard form
'
+'- Factor the polynomial into a product of linear factors
'
+'- Use the Zero Factor Property to set each factor equal to zero
'
+'- Solve each of the resulting linear equations.
'
+'
'
+'The resulting solutions are solutions of the original equation.';
var zero_factor_property =
'If a and b are real numbers and ab = 0, then a = 0 or b = 0.';
var zero_quotient_property =
'A fraction is equal to zero if and only if its numerator is zero.';
var polynomial_equation_solving_by_factoring =
'It is sometimes possible to use factoring in conjunction with the Zero Factor Property to solve a '
+'polynomial equation.';
var polynomial_equation_solving_by_substitution =
'It is sometimes possible to substitute a variable for an expression in an equation so that the resulting '
+'equation is a quadratic equation. That quadratic equation may be solved by normal methods for solving quadratic '
+'equations. Those results are then substituted back to obtain solutions to the original equation.';
var function_notation_comment1 =
'The symbol f(x) is read: f of x
'
+'Note that the element inside the parenthesis is a single element of the domain
'
+'Note that f(x) is a single element of the range of the function.';
var function_notation_comment2 =
' In reference to the notation above, it is correct to speak of the function f.
'
+'In reference to the notation above it is incorrect to speak of the function f(x).';
var function_notation_comment3 =
' Function as an Equation: Function notation is most commonly used to state the rule of a function. This use of function notation'
+' is especially convenient when the rule for the function can be expressed as an equation in terms of the '
+'domain element. ';
var function_notation_convention1 =
'Common practice is to write only the rule for a function with no mention of the domain or range. '
+'To avoid confusion the following convention has been adopted. Unless otherwise stated, the domain of '
+'a function is the largest set of real numbers for which the rule makes sense (has meaning) and the range '
+'is the set of real numbers associated with those domain elements';
var evaluating_a_funtion=
'The expression evaluating a function is an abreviation for evaluating a '
+'function at a given point which means to determine the range value associated '
+'with a given domain element. To evaluate a function then means to calulate f(x) '
+'for a given value of x. This is particularly easy to do if the rule for the '
+'function can be expressed as an equation and function notation is used to express '
+'that equation.';
var function_calulating_range_values =
'To calculate the unique range value associated with a domain element one must follow exactly the recipe '
+'given in the rule of the function. That means you must perform all operations on the domain element '
+'which the rule dictates. Remember the domain element is the entire expression inside the parenthesis.';
var finding_zeros_of_a_function =
'To find the zeros of a function we must recall the definition of zero of a function. '
+'That definition tells us to find domain elements x whose corresponding range element f(x) is zero.
'
+'Therefore to find the zeros of a function f, we must
first solve the equation resulting from f(x) = 0
'
+' and secondly, discard those solutions which are
not in the domain of the function';
var finding_zeros_of_a_rational_function =
'To find the zeros of a rational function we use the Zero Quotient Property which states that a fraction is 0 '
+'if and only if its numerator is 0 and its denominator is not 0.
This implies that we must
find the zeros of the function in the numerator.
'
+'Zeros of a function are domain elements so we must discard all zeros of the numerator which are not in the domain of the function f. '
+'Therefore we must
determine the domain of f. ';
var possible_rational_zeros =
'TO BE DELETED If p/q is a rational zero of a polynomial function with integer coefficients,
'
+' then p must be a divisor of the constant term and q must be a divisor of the leading coeficient.'
var convention_for_the_domain =
'Unless otherwise stated the domain of a function is the largest set of real numbers for which the rule '
+'makes sense. ';
var domain_of_polynomial_function =
'Unless otherwise stated the domain of a function is the largest set of real numbers for which the rule '
+'makes sense.
Therefore the domain of a polynomial function, unless otherwise stipulated, '
+'is the set of all real numbers . ';
var domain_of_rational_function =
'Unless otherwise stated the domain of a function is the largest set of real numbers for which the rule '
+'makes sense.
Therefore the domain of a rational function, unless otherwise stipulated, '
+'is the set of all real numbers for which the denominator is not zero . ';
var domain_of_function_with_radical =
'Unless otherwise stated the domain of a function is the largest set of real numbers for which the rule '
+'makes sense.
Therefore the domain of a polynomial function which contains a radical, '
+'unless otherwise stipulated, is the set of all real numbers for which the expression under the radical is . '
+'non-negative.';
var linear_function_vertical_line =
'A vertical line is not the graph of a function.';
var linear_function_graph =
'The graph of a linear function is a non-vertical line and
conversely every non-vertical line '
+'is the graph of a linear function.';
var linear_function_simple_concept =
'Because lines and linear functions are simple concepts, there are relatively few questions one can ask about them.';
var linear_function_nine_questions =
'When given a linear function:
'
+' '
+' - Write the rule for the function in the form f(x) = mx + b.
'
+' - Sketch the graph of f.
'
+' - What is the y-intercept of the graph of f ?
'
+' - What is the zero of f ?
'
+' - What is the x-intercept of the graph of f ?
'
+' - Is f increasing or decreasing ?
'
+' When asked to determine a linear function:
'
+' - Determine the rule for the linear function whose graph has a given slope and a given y-intercept.
'
+' - Determine the rule for the linear function whose graph has a given slope and contains a given point.
'
+' - Determine the rule for the linear function whose graph contains two given points.
'
+'
';
var linear_equations_eight_questions =
'When given a linear equation in two variables:
'
+' '
+' - Write the equation in two variables in the form y = mx + b.
'
+' - Sketch the graph of the equation.
'
+' - What is the y-intercept of the graph of the equation ?
'
+' - What is the x-intercept of the graph of the equation ?
'
+' - Is the graph increasing or decreasing ?
'
+' When asked to determine a linear equation in two variables:
'
+' - Determine the equation for the linear equation in two variables whose graph has a given slope and a given y-intercept.
'
+' - Determine the equation for the linear equation in two variables whose graph has a given slope and contains a given point.
'
+' - Determine the equation for the linear equation in two variables whose graph contains two given points.
'
+'
';
var looking_for_a_linear_function =
'To determine a particular linear function use the fact that the rule of the desired function can be written '
+'in the form
f(x) = mx + bwhere m is the slope of the graph
and b is the y-intercept of the graph.
'
+'Then use other information from the problem to determine values for the coefficients m and b '
+'for the desired function.';
var linear_function_determine =
'The rule of a linear function can be written in the form
f(x) = mx + b'
+'where m is the slope of the graph
and b is the y-intercept of the graph.
'
+'To determine'
+' a particular linear function means to determine values for both of the coefficients m and b. ';
var linear_function_partially_determine =
'The rule of a linear function can be written in the form
f(x) = mx + b'
+'where m is the slope of the graph
and b is the y-intercept of the graph.
'
+'To partially determine a particular linear function means to determine values '
+'for one of the coefficients m and b. ';
var linear_function_graphing =
'To graph a linear function it is only necessary to plot two points on the graph and draw the line through those '
+'points.
The x-intercept is always important so it should be plotted and labeled.
The y-intercept is easy to '
+'determine, so it is a good point to plot and label.';
var linear_equation_two_variables_graphing =
'';
var linear_inequality_two_variables_graphing =
'
'
+' 1. This means that if both sides of an equation are squared and the resulting equation is solved, '
+'those solutions are the only candidates as solutions to the original equation.
'
+' 2. It also means that if both sides of an equation are squared and the resulting equation is solved, '
+'some of those solutions may not be solutions to the original equation.';
var radical_equation_three =
'To solve equations involving radicals, it is necessary to clear the equation of radicals. This is usually done by '
+'squaring both sides of the equation. The resulting equation may then be solved to obtain a solution set which '
+'contains the original solution set. Therefore an integral part of the solution process is to test each of the '
+'possible solutions in the original equation.';
var radical_equation_four =
'When squaring both sides of an equation is part of the solution process, testing all the possible solutions '
+'in the original equation must also be a part of the solution process.';
var rational_equation_one =
'When both sides of an equation are multiplied by an expression containing a variable there is no assurance that '
+'the resulting equation will be equivalent to the original. Consequently it must be assumed that the resulting '
+'equation is not equivalent to the original equation.';
var rational_equation_two =
'When both sides of an equation are multiplied by an expression containing a variable the solution set of the '
+'resulting equation contains the solution set of the original equation.
'
+' 1. This means that if both sides of an equation are multiplied by an expression containing a '
+'variable and the resulting equation is solved, those solutions are the only candidates as solutions to the '
+'original equation.
'
+' 2. It also means that if both sides of an equation are multiplied by an expression containing '
+'a variable and the resulting equation is solved, some of those solutions may not be solutions to the original '
+'equation.';
var rational_equation_three =
'Therefore when multiplying both sides of an equation by an expression containing a variable is part of the '
+'solution process, testing all the possible solutions in the original equation must also be a part of the solution '
+'process.';
var rational_equation_four =
'To solve equations containing rational expressions, clear the equation of fractions by multiplying both sides of '
+'the equation by the LCD of all rational expressions. Then solve as usual. Finally check each of these possible '
+'solutions in the original equation to determine its solution set.';
var long_division_natural_numbers =
' Long division is a process for calculating the quotient q and remainder r whose existence is assured by the '
+'Division Algorithm.
Long division is also used to calculate the decimal equivalent of a fraction.';
var long_division_polynomials =
' Long division and Synthetic division are processes for calculating the quotient q and remainder r whose existence is assured by the '
+'Division Algorithm for Polynomials.';
var remainder_theorem =
'If a polynomial function f is divided by the polynomial function whose rule is g(x) = x - k, '
+'then the remainder is f(k).';
var factor_theorem =
'If g is a polynomial function whose rule is g(x) = x – k then g is a factor of a polynomial function f '
+'if and only if f(k) = 0.';
var factor_theorem_fact1 =
'The above statement means:
'
+''
+'- If g(x) = x – k and g is a factor of the polynomial function f, then f(k) = 0.
'
+'- If f(k) = 0 for a polynomial function f, then the function defined by g(x) = x – k is a factor of f.
'
+'
';
var factor_theorem_fact2 =
'The complex number k is a zero of the polynomial function f if and only if the function whose rule is '
+'g(x) = x – k is a factor of f.';
var possible_rational_zeros =
'If 
is a rational zero of a polynomial function with integer coefficients,
'
+' then the numerator p must be a divisor of the constant term and the denominator q must be a divisor of '
+'the leading coeficient.';
var fundamental_theorem_of_algebra =
'If f is a polynomial function of degree n > 0, then f has at least one complex zero.';
var complex_zeros1 =
'If f is a polynomial function of degree n > 0 then f has exactly n complex zeros.';
var complex_zeros2 =
'Suppose f is a polynomial function with real coefficients.
If the complex number a + bi is a zero of '
+'the function f, then its conjugate a - bi is also a zero of the function f.
'
+'This fact is usually remembered by observing that complex zeros of polynomial functions occur in conjugate pairs.';
var linear_and_quadratic_factors =
'Every polynomial of degree n > 0 with real coefficients can be written as the product of linear and quadratic '
+'factors with real coefficients, where the quadratic factors have no linear factors with real coefficients.';
var prime_polynomials =
'Consider the set of all polynomials together with the usual operations of addition and multiplication.
'
+'In that set linear polynomials with real coefficients are prime polynomials.
'
+'In that set quadratic polynomials with '
+'no linear factors with real coefficients are prime polynomials.';
var prime_factorization_of_polynomials =
'In view of this observation the previous fact may be restated as follows:
'
+'Every polynomial of degree n > 0 with real coefficients can be written as the product of prime polynomials.';
var rational_function_domain_convention =
'As with all functions the convention for the domain of a rational function is all real numbers for which '
+'the rule is defined ( makes sense). Therefore the domain of a rational function is all real numbers except the '
+'zeros of the denominator.';
// Start SYSTEMS OF EQUATIONS AND INEQUALITIES
var graph_of_linear_equation_in_two_variables =
'The solution set for a linear equation in two variables is the set of ordered pairs '
+'of numbers which make the equation true. The graph of a linear equation in two variables is the line '
+'whose coordinates are solutions of the equation. The graph is a picture of the solution set.';
var graph_of_system_of_two_linear_equations_two_variables =
'The solution of a system of two equations in two variables is the pair (or pairs) '
+'of numbers that make both equations true. The graph is therefore the point (or points) '
+'which is (are) on both lines. Therefore the graph of the solution of a system of two '
+'equations in two variables is the point (or points) of intersection of the two lines.';
var system_of_equations_replacement =
'In a system of linear equations, replacement of an equation with an equivalent equation '
+'produces a system which is equivalent to the original system.
'
+' '
+'(A) If the same expression is added to (or subtracted from) '
+'both sides of an equation the resulting equation will be equivalent to the original '
+'equation.
'
+' '
+'(B)If both sides of an equation are multiplied ( or divided) '
+'by the same non-zero real number, the resulting equation is equivalent to the original '
+'equation.';
var system_of_equations_substitution =
'In a system of linear equations, if the value of one of the variables is known, an '
+'equivalent system is generated if that value is substituted into the equations.';
var system_of_equations_linear_operation =
'In a system of linear equations, if two equations are added (or subtracted) and one '
+'(but not both) of the summand equations is replaced with the sum, the resulting system '
+'of equations is equivalent to the original system of equations.';
var determined_variable =
'If a linear equation in a system of equations has only one variable, then the value of '
+'that variable is determined for the system.';
var graph_of_sum_of_equations =
'If two linear equations in two variables are added, the graph of the third equation '
+'(the sum) intersects the two original graphs at their intersection point. '
+'Moreover, if the original two graphs are parallel, the third will be parallel to the '
+'original graphs.';
var basis_for_solution_methods =
'Every method for solving a system of linear equations is based on the above methods '
+'for generating equivalent systems of equations.';
var solution_is_an_ordered_pair =
'It is important to realize that a solution to a system of equations in two variables is an ordered '
+'pair of real numbers. It is not two numbers nor is it an ordinary set of two numbers.';
var substitution_method_two_equations_two_variables =
'The following method, called the Substitution Method, is suitable for solving a system '
+'of two equations in two variables and some special larger systems.'
+''
+'- Solve for one of the variables in one of the equations.
'
+'- Substitute that value into the other equation to obtain an equation in one variable.
'
+'- Solve that equation for the single variable.
'
+'- Substitute that value into the equation in Step 1 and solve for the variable.
'
+'- The obvious ordered pair is solution of the system.
'
+'
';
//INDEX_MATRICES
var representing_matrices =
'In general we use capital letters to represent matrices and lower case letters to '
+'represent entries of a matrix. We use subscripts on the entries to indicate the row '
+'and column which the entry occupies.
'
+'
';
var matrix_sum_different_order =
'The sum of two matrices with different orders is not defined.';
var matrix_sum_opposite =
'The sum of a matrix and its opposite is the zero matrix.';
var matrix_addition_properties =
'If A, B, and C are matrices with the same orders and both c and d are scalars, then'
+''
+'1. Matrix addition is commutative
'
+'A + B = B + A
'
+'2. Matrix addition is associative
'
+'A + (B + C) = (A + B) + C
'
+'3. Scalar Multiplication is associative
'
+'(cd)A = c(dA)
'
+'4. The number 1 is the scalar identity
'
+'1A = A
'
+'5. Scalar multiplication distributes over matrix addition
'
+'c(A + B) = cA + cB
'
+'6. Scalar addition distributes over scalar multiplication
'
+'(c + d)A = cA + dA
'
+'7. The zero matrix O is the additive identity
'
+'A + O = O + A = A
'
+'
';
var matrix_multiplication_properties =
'If A, B, and C are matricies and c is a scalar, then'
+''
+'1. Matrix multiplication is associative
'
+'A(BC) = (AB)C
'
+'2. Matrix multiplication on the left distributes over matrix addition
'
+'A(B + C) = AB + AC
'
+'3. Matrix multiplication on the right distributes over matrix addition
'
+'(A + B)C = AC + BC
'
+'4. Multiplicative Identity
'
+'If A is an n×n matrix and In is the identity of order n, then
'
+'InA = AIn = A
'
+'5. Scalar multiplication is distributive with '
+'respect to matrix multiplication
'
+'c(A + B) = cA + cB
'
+'
';
var matrix_inner_product_remember =
'An easy way to remember the definition of inner product is to observe that '
+'the inner product is simply the sum of the products of corresponding entries.';
var matrix_inner_product_example =
'The following illustrates computing the inner product of a 1×3 row matrix with '
+'a 3×1 column matrix.
'
+'
';
var matrix_inner_product_scalar =
'It is important to remember that the inner product of two matrices does not yield a matrix, '
+'but yields a real number and those are called scalars in the context of matrices.';
var matrix_product_dimension_factors =
'The product of two matrices is defined only if the number of columns of the first '
+'is equal to the number of rows of the second. The order of the product is the '
+'number of rows of the first times the number of columns of the second. An easy '
+'way to keep track of these conditions is to write the dimensions of the matrices'
+' to be multiplied and then let the following diagram be your guide.
'
+'
';
var matrix_product_ij_entry =
'The entry in the ij position of product matrix is the inner product of the '
+'ith row of the first matrix and the jth column of the '
+'second matrix. For example, the entry in the 23 position (second row, third column)'
+'is calculated by computing the inner product of the second row of the first '
+'matrix times the third column of the second matrix. Perhaps the following '
+'colored diagram will help.'
+'
';
var matrix_multiplication_schematic =
'The following schematic diagram shows how to determine the rows and columns to use '
+'in the various inner products when computing the product of two matrices. '
+'Pay attention to the indices of the positions in the product matrix.
'
+'
';
var matrix_multiplication_explained =
'The above definition is hard to understand because of the subscripting. Matrix '
+'multiplication is not as complicated as it looks. Each entry in the product matrix '
+'is the inner product of a row from the first matrix and a column from the second matrix.'
+' The index on the entry in the product tells you which row and column to use. For '
+'example to compute the entry in the 35 position '
+'(3rd row 5th column), use the third row and fifth column.';
var matrix_multiplication_example =
'A simple numeric example should cement your understanding of matrix multiplication.
'
+'
';
var matrix_multiplication_not_commutative =
'It is important to recognize and remember that matrix multiplication is not commutative.'
+' Generally BA ≠ AB, although there are a few exceptions. Unless verified otherwise it '
+'must be assumed that BA is not equal to AB. The following multiplications yield an example to show '
+'that matrix multiplication is not commutative.';
var matrix_multiplication_not_commutative_example =
'
';
var matrix_example1 =
'When it is desirable to emphasize the notation used for individual entries we use '
+'[aij] instead of capital A to name the matrix.'
+'
'
+'
and at other times we simply use a rectangular array to represent the matrix.'
+'
';
var matrix_sum_example =
'The following illustrates the calculations required to compute the sum of two '
+'3×3 matrices.
'
+'
';
var matrix_scalar_product_example =
'The following illustrates the calculations required to compute a scalar product.
'
+'
';
var det_notation =
'The name of the determinant function is det. Vertical bars '
+'(such as used in absolute value) are also used to denote the determinant function. '
+'For example if A is a matrix whose determinant is 5, we would write '
+'det(A) = 5 or | A | = 5.';
var det_function_notation =
'Because the determinant is a function we will emphasize function notation and in most '
+'references we will use its name det. '
+'The normal function notation as used with the function named det is:
'
+'
';
var det_function_notation_extension =
'Historical use of notation, and sometimes convenience, requires a knowledge of and an '
+'ability to use vertical bars to designate the determinant of a matrix. Therefore it '
+'is common to encounter some notational variations.
'
+'Instead of det(A) we may use |A|.
'
+'Similar dual symbolism was encountered with the abs function and vertical bars for absolute value.
'
+'When we want to emphasize the entries of the matrix we normally will write
'
+'
'
+'or we could use the vertical bar notation
'
+'
'
+'In summary:
'
+'
';
var det_function_general_comments =
'Functions normally studied prior to this have had some common characteristics which are '
+'not shared by the det function.
'
+' The domain of each of the earlier functions has been some subset of the real numbers.
'
+' The domain of det is the set of square matrices.
'
+'The rule for most of the earlier functions could be expressed as an equation '
+'(or a couple of equations).
'
+'The rule for the det function is a recursive rule and cannot be expressed as an equation.
'
+'The meaning of recursive will become clear very shortly.';
var det_function_defined_recursively =
'The following procedure will be used to describe the rule for the det function:'
+''
+'- Describe the rule for the det of a 2×2 matrix in terms of its entries.
'
+'- Describe the rule for the det of a 3×3 matrix in terms of the det of a 2×2 matrix.
'
+'- Describe the rule for the det of a 4×4 matrix in terms of the det of a 3×3 matrix.
'
+'- Describe the rule for the det of a 5×5 matrix in terms of the det of a 4×4 matrix.
and in general '
+'- Describe the rule for the det of a n×n matrix in terms of the det of a (n-1)×(n-1) matrix.
'
+'
'
+'The technique illustrated with this definition; namely to define each item in terms '
+'of its predessor in the sequence is called recursion and the definition is said to be '
+'a recursive definition.';
var det_example_2by2 =
'
';
var det_example_ahat =
'
';
var det_example_cofactor =
'
'
+'
';
var det_explanation_nbyn =
'The rule may seem a bit confusing at first but is not really very complicated.
'
+'Note that a11 is the entry in the 11 position of the matrix and A11 is its cofactor.
'
+'Note that a12 is the entry in the 12 position of the matrix and A12 is its cofactor.
'
+'and so on across the first row of the matrix A
'
+'Note that a1n is the entry in the 1n position of the matrix and A1n is its cofactor.
'
+'
So det(A) is just the sum of the products of each entry times its cofactor in the first row.';
var det_amazing_row_col_expansion =
'An amazing thing about the det function is that one can use any row to calculate '
+'the determinant and the value will always be the same for a given matrix.
'
+'The really amazing thing is that one can use any column to calculate det(A).';
var det_row1_expansion_general_illustration_3by3 =
'The computations here with a general 3×3 matrix should help understand the '
+'rule for the function det.
The cofactor expansion along the first row will '
+'be illustrated. Pay close attention to the subscripts and exponents. Subscripts for '
+'an entry remain the same throughout the computations. The subscripts always represent '
+'the entry′s position in the original matrix. Note that the exponent on (-1) '
+'is the sum of the indices of the matrix entry.'
+'
';
var det_row1_expansion_example =
'This illustration uses cofactor expansion along Row 1.'
+'
';
var det_row2_expansion_example =
'This illustration uses cofactor expansion along Row 2.'
+'
'
+'
Notice that if an entry in the matrix is 0 then expansion along that row decreases the '
+'computation.';
var det_row3_expansion_example =
'This illustration uses cofactor expansion along Row 3.'
+'
';
var det_of_product =
'If A and B are square matrices with the same dimension, then '
+'det(AB) = det(A)det(B).';
var det_is_zero =
'If A is a square matrix and det(A) = 0, then A does not '
+'have an inverse.';
var det_applications =
'There are many other properties of the function det. A few of the many applications '
+'of the determinant function are mentioned here.'
+''
+'- The function det can be used to solve systems of equations. This generally involves Cramer′s rule.
'
+'-
'
+'- If the vertices of a triangle are (x1,y1), '
+'(x2,y2), (x3,y3) '
+'then its area A is given by the formula:
'
+'
'
+' '
+'-
'
+'- Three points are colinear if and only if the area of the triangle with the three points '
+'as its vertices has area zero. So a test for colinearity is to see if the equation '
+'
'
+'
is true. '
+'-
'
+'- The following det equation is the equation of the line through the points '
+'(x1,y1) and (x2,y2).
'
+'
'
+' '
+'
';
.';
var quadratic_function_vertex_location_of =
'There are several convenient ways to locate the vertex of the graph of a quadratic function:'
+'
:
is a horizontal asymptote.
'
+'are symbols for division.
'
+'are used to indicate multiplication. When there is no chance of confusion, multiplication '
+'is indicated by simple juxtaposition.';
var multiplication_symbols =
'The symbols 
';
var function_inverse_comment9 =
'Replacing f(x) with a single variable y is for convenience only. The entire process can be carried out without '
+'such replacement, but the notation becomes awkward.';
var function_inverse_comment10 =
'It is INCORRECT to speak of an inverse function. Just as it is incorrect to speak of 3/4 as an inverse.
or 
';
var factoring_by_grouping =
'To factor a polynomial with four or more terms a technique called '
+'Factoring by Grouping is sometimes productive. '
+' After factoring out the GCF the procedure is as follows:'
+'
to find the midpoint of the '
+' given diameter.
'
+'in the more familiar form '
+'
';
var comment_exercise_1_1_56 =
'In case you are wondering how I got 17. Notice that as a result of the definition of 
';
var finding_x_intercept =
'A point is on the x-axis if and only if its second coordinate y is zero.
.';
var conjugate_of_a_real_number =
'A real number is its own conjugate
.