Graphing Linear Equations

What is a linear equation? Any equation that can be written

as Ax+By=C where A and B are both not 0.

Ax+By=C is called the general or standard form of the equation of a line.

Examples of linear equations 2x-3y=6

3x=1-2y

Examples of linear equations y=-2x+3

(1/2)x=.75y-3

Examples of non-linear equations

2xy

3xy

Examples of non-linear equations

xy

xy

Ways to Graph Linear Equations

Table of Values

x and y intercepts

Slope Intercept Form: y = mx + b

Table of Values

x y

Table of Values You can select any value for x OR

y, then you can solve for the other variable.

If you choose a value for x, then input that value into the linear equation and solve for y. If you choose a y value, then solve for x.

Table of Values To create a line you need 2

coordinates.

To be safe find 3 coordinates. If one point is incorrect, you will know because the coordinates will not create a straight line.

Table of Values Example: Given: 2x + 3y = 6 Solve:

x y

01

-2

Table of Values When x = 0 2(0) + 3y = 6 2 + 3y = 6 -2 -2 3y = 6 3 3 y = 2

x y

01

-2

2

Table of Values When y = 1 2x + 3(1) = 6 2x + 3 = 6 - 3 - 3 2x = 3 2 2 x = 3/2

x y

01

-2

23/2

Table of Values When x = -2 2(-2) + 3y = 6 -4 + 3y = 6 +4 +4 3y = 10 3 3 Y = 10/3

x y

01

-2

23/2

10/3

Table of Values List the points

(0, 2) (3/2, 1) = (1.5, 1) (-2, 10/3) = (-2, 3

1/3)

x y

01

-2

23/2

10/3

Table of ValuesPlot the points(0, 2)(1.5, 1) (-2, 3 1/3)

Table of Values Draw a line

through the points

You have graphed the equation:

2x + 3y = 6

Graph by x and y intercepts The x-intercept of a line is the

point (a,0) where the line intersects the x-axis. To find a, substitute 0 for the y and solve for x.

The y-intercept of a line is the point (0,b) where the line intersects the y-axis. To find b, substitute 0 for the x and solve for y.

Graph by x and y intercepts Example:

Graph 2x +3y = 6 by finding the x and y intercepts.

Graph by x and y intercepts Find the x-intercept by making y = 0.

2x – 3(0) = 6 2x = 6 2 2 x = 3 x-intercept = (3, 0)

Graph by x and y intercepts Find the y-intercept by making x = 0.

2(0) + 3y = 6 3y = 6 3 3 y = 2 y-intercept = (0, 2)

Graph by x and y intercepts Plot the: x-intercept (3, 0) y-intercept (0, 2)

Graph by x and y intercepts Draw a line

through the intercepts and you have graphed:

2x + 3y = 6

Slope intercept form:y = mx + b The slope intercept form is y = mx

+ b where “m” is the slope and “b” is the y- intercept.

The x and y represent the coordinates which satisfy the linear equation.

Slope intercept form:y = mx + bThe slope is all of the following: rate of change of a linear graph. change in y over the change in x. vertical change over the

horizontal change. rise over run.

Slope intercept form:y = mx + b Different Types of slopes

Positive Negative Slope of zero Slope is undefined

Slope intercept form:y = mx + b Examples of:

Positive Slopes

Slope intercept form:y = mx + b Examples of:

Negative Slopes

Slope intercept form:y = mx + b Examples of: Slopes of zero

Slope intercept form:y = mx + b Examples of: Slopes is undefined

Slope intercept form:y = mx + b The slope formula is used to

find the slope given two coordinates.

Slope formula:

12

12

xxyym

Slope intercept form:y = mx + b Example:

Find the slope given:

(2, -3) and (-4, -1)

Slope intercept form:y = mx + b First label the coordinates:

2211

)1,4()3,2(yxyx

Slope intercept form:y = mx + b Substitute into

the slope formula and solve.

12

12

xxyym

41

82

)2(4)3(1

m

Slope intercept form:y = mx + b

Slope can be determined by calling it rise/run.

The rise determines if you go up or down.

If the number is (+), then go up If the number is (-), then go down.

Slope intercept form:y = mx + b The run determines if you go left

or right. If the number is (+), then go to the

right. If the number is (-), then go to the

left.

Slope intercept form:y = mx + b In our example, we got a slope of: (-1/4) The rise is: -1 The run is: 4 This means from a point on the

graph, I would go down 1 and to the right 4 to find another point on the graph.

Slope intercept form:y = mx + b To graph using the slope intercept

form: the equation must be in slope intercept

form you must determine your “m” and “b” plot your “b” use “m” to find another point then draw a line through those

coordinates

Slope intercept form:y = mx + b Example:

Graph: 2x +3y = 6 using slope intercept form

Slope intercept form:y = mx + b

Step 1: Convert to slope intercept form

2x + 3y = 6 -2x -2x 3y = -2x + 6 3 3 y = (-2/3) x + 2

Slope intercept form:y = mx + b Step 2: Determine your “m” and

“b”

y = (-2/3) x + 2 m = (-2/3) b = 2

Slope intercept form:y = mx + b Step 3: Plot your “b” (y-intercept)

b = 2

Slope intercept form:y = mx + b Step 4: Use “m” to find another

point m = (-2/3) rise = -2 and the run = 3 From the “b” (y-intercept) you

would go down 2 Then go to the right 3

Slope intercept form:y = mx + b After you use the

“m” to find the other point, then you draw a line through the 2 points and you have graphed 2x+3y=6.

Now that you are done:“Try these practice problems using the various methods.”

4214332

yxxyxy

y= -2x + 3

3y = -4x - 1

x – 2y = 4