Important Properties of Linear Equations

 

Definition:  A linear equation in two variables is an equation which may be written in the form

y = mx + b    where m, and b are real numbers.

 

The graph of a linear equation is a non-vertical  line with slope m and y-intercept b.

 

Every non-vertical line is the graph of a linear equation of the form y = mx + b

 

The x-intercept occurs when y = 0. 

Therefore we find the x-intercept by solving mx + b = 0. 

Subtract b from both sides and then divide both sides by m to obtain .

 

The y-intercept occurs when x = 0. 

Therefore the y-intercept is  b. 

 

To sketch the graph of a linear equation, plot any two points whose coordinates satisfy the equation and draw the line passing through the two points.  Plotting the two intercepts is generally a good idea.

 

Any time two independent pieces of information are known about a line L it is possible to determine the linear equation whose graph is that line L.  The most important fact used in this process is:

 A point is on the graph of an equation if and only if its coordinates satisfy the equation. 

That is, a point (t, k) is on the graph of an equation f if and only if substituting t and k into the equation yields a true statement.

 

How Does This Work?

 

Case 1:  If  the slope and y-intercept are known, simply replace m and b in y = mx + b with the known values.

 

Case 2:  If the slope and a point are known, replace m in

y = mx + b with the known value for m.  Then use the fact that the point is on the graph to obtain an equation with b as the only unknown.  Solve for b and substitute it into the partially determined equation.

 

Case 3:  If two distinct points are known, calculate the slope of the line segment joining the two points, select one of the points and revert to Case 2.

 

Example of Case 2:  Find the linear equation whose graph has slope 3 and passes through the point (4, -3).

Solution:  Since the desired equation is linear, it has the form

y = mx + b.

Since its slope is 3, it has the form y = 3x + b.  (*)

Since (4, -3) is on the graph, its coordinates satisfy the rule of the equation.

That means a true statement results when 4 and –3 are substituted into (*)

Using (*) we get -3 = (3)(4) + b

Solving for b yields b = -15.

Substitute that value into the partially determined equation given in (*) to obtain

y = 3x - 15  as the linear equation whose graph is the line with slope 3 and passes through the point (4, -3).

 

Alternate methods for determining the equation of  a line depend on remembering formulas for each of the cases.

 

If the slope and y-intercept are known, use the

Slope Intercept Form of the Equation of a line

 y = mx + b

 

If the slope and one point (x1, y1) on the line are known use the

Point-Slope Form of the equation of a line

y – y1 = m(x – x1)

 

If two points (x1, y1) and (x2, y2) are known to be on the line, then use the

Two Point Form of the equation of a Line

 

           

 

General Form for the equation of a line

 Ax + By = C where A, B, and C are real numbers and not both A and B are zero.

 

The Equation of a Vertical Line

has the form x = a, where a is the x-intercept of the vertical line.

Definition:  The slope of the non-vertical line through two points (x1, y1) and (x2, y2) is 

 

                

 

Fact:  The midpoint of the line segment joining two points (x1, y1) and (x2, y2) is the point 

             

                     

 

Fact:  The distance between two points

(x1, y1) and (x2, y2) is

 

             

 

Fact:  Two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals of each other.  The statement that they are negative reciprocals of each other may be stated algebraically with any one of the following equations.

   

   

 

 

Fact:  Two non-vertical lines are parallel if and only if they have different y-intercepts and they have the same slopes.