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

Examples of non-linear equations

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 1 -2

Table of Values x y When x = 0 2(0) + 3y = 6 2 + 3y = 6 -2 -2 3y = 6 -2 -2 3y = 6 3 3 y = 2 x y 1 -2 2

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

Table of Values x y When x = -2 2(-2) + 3y = 6 -4 + 3y = 6 +4 +4 +4 +4 3y = 10 3 3 Y = 10/3 x y 1 -2 2 3/2 10/3

Table of Values x y List the points (0, 2) (3/2, 1) = (1.5, 1) (-2, 10/3) = (-2, 3 1/3) x y 1 -2 2 3/2 10/3

Table of Values Plot 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 + b The 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:

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:

Slope intercept form: y = mx + b Substitute into the slope formula and solve.

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.”

y= -2x + 3

3y = -4x - 1

x – 2y = 4