Linear Functions

6.6 : General Form of the Equation for a Linear Relation

Today’s Objectives

• Relate linear relations expressed in: slope-intercept form, general form, and slope-point form to their graphs, including:• Express a linear relation in different forms, and

compare graphs• Rewrite a linear relation in either slope-intercept or

general form• Graph, with or without technology, a linear relation in

slope-intercept, general, or slope-point form• Identify equivalent linear relations from a set of linear

relations• Match a set of linear relations to their graphs

Equations of a Linear Function

• We can make an equation that describes a line’s location on a graph. This is called a linear equation. There are three forms of linear equation that we will be looking at:• Standard Form: Ax + By + C = 0, where A, B, and

C are integers.• Slope y-intercept form: y = mx + b, where m is

the slope, and b is the y-intercept. • Slope-point form: y – y1 = m(x – x1), where m is

the slope, and the line passes through a point located at (x1, y1)

General form (or standard form)

• Another form for the equation of a linear function is general form, or standard form:

• In certain situations you will be asked to change the equation from slope y-intercept form into general form, and vice versa. • When converting the equation into standard

form, we must remember that A, B, and C, MUST be integers, which means all fractions need to be removed. We also need to remember to move all terms to the same side of the equation to make it equal to zero.

ExampleWrite the following in general form:

y – 1 = 2/3(x + 2)

Multiply both sides by 3: 3(y – 1) = 2(x+2)

Expand: 3y – 3 = 2x + 4

Make equation = 0 3y – 2x – 7 = 0Collect Like TermsArrange in proper order -2x + 3y – 7 = 0

ExampleWrite the following in general form:

y + 2 = 3/5(x - 4)

Multiply both sides by 55y + 10 = 3(x – 4)

Remove the brackets5y + 10 = 3x – 12

Collect all terms on one side of the equation5y = 3x – 12 – 100 = 3x – 5y - 22

What does the graph look like?A=0 Ax + By + C = 0

By + C = 0y = -C/B

Since –C/B is a constant, the graph will be a horizontal line

B=0 Ax + By + C = 0Ax + C = 0x = -C/A

Since –C/A is a constant, the graph will be a vertical line

Example

Graph the line 2x – 5y – 10 = 0

Determine the x-intercept Determine the y-interceptLet y=0 and solve for x Let x=0 and solve for y2x – 5(0) – 10 = 0 2(0) – 5y – 10 = 02x – 10 = 0 -5y – 10 = 02x = 10 -5y = 10x = 5 y = -2(5,0) (0,-2)

Plot the points and draw a line through them.

ExampleDetermine the slope of 2x – 5y – 10 = 0

Rewrite the equation in slope-interval form:

2x – 5y – 10 = 0 solve for y-5y = -2x + 10y = -2x + 10

-5

y = -2x/-5 + 10/-5

y = 2/5x – 2

m = 2/5

ExampleDetermine the slope of -3x – 2y – 16 = 0

Rewrite the equation in slope-interval form:

-3x – 2y – 16 = 0 solve for y-2y = 3x + 16y = 3x + 16

-2

y = 3x/-2 + 16/-2

y = -3/2x – 8

m = -3/2