DrDelMath

SUMMARY
Chapter 6:  Systems of Equations and Inequalities

Section 6.1: Solving Systems of Equations

DEFINITION: A solution of a system of linear equations is an ordered n-tuple of numbers which makes all of the equations true when the n-tuple of numbers is substituted into the equations.
DEFINITION: Two systems of equations are equivalent systems if they have the same solutions.

PROCESS: METHODS FOR GENERATING EQUIVALENT SYSTEMS
1) (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.

2) (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.

3) (Linear Operations) 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.

FACT: If a linear equation in a system of equations has only one variable, then the value of that variable is determined for the system.
FACT: If two linear equations in two variables are added, the graph of the third equation (the sum) intersects the two original graphs at their original intersection point. Moreover, if the original two graphs are parallel, the third will be parallel to the original graphs.

Section 6.2: Two-Variable Linear Systems

Section 6.3: Multivariable Linear Systems

Section 6.4: Systems of Inequalities

Definition: A linear inequality in two variables is an inequality that can be written in one of the following four forms:

Definition: A point (x, y) is a solution of an inequality in two variables if the coordinates satisfy the inequality. (that is, if a true statement results when the coordinates are substituted for the variables in the inequality.)

Definition: The graph of an inequality is the set of points which are solutions of the inequality. (That is, the graph is the set of all points whose coordinates satisfy the inequality).

Definition: If the inequality symbol in an inequality in two variables is replaced with an equality symbol, the graph of the resulting equation is called the boundary line for the inequality.

FACT: The graph of an inequality in two variables is a half-plane.
FACT: If the inequality symbol is replaced with an equal symbol, its graph forms the boundary between the half-plane consisting of all solutions of the inequality and the half-plane consisting of all points which are not solutions of the inequality.

PROCESS: To graph an linear inequality in two variables:

a) Sketch the graph of the boundary line

i) as a solid line if the inequality symbol is either or .
ii) as a dashed line if the inequality symbol is either > or <.

b) Pick a point, not on the boundary line, as a test point and substitute its coordinates into the inequality.
c) If the result from Step b is a TRUE statement, the half-plane containing the test point is the solution.
d) If the result from Step b is a FALSE statement, the half-plane which does not contain the test point is the solution.
e) Shade the half-plane which is the solution and label all important points.