1. Graphing Linear Equations

2. 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.

3. Examples of linear equations
2x-3y=6
3x=1-2y

4. Examples of linear equations
y=-2x+3
(1/2)x=.75y-3

5. Examples of non-linear equations

6. Examples of non-linear equations

7. Ways to Graph Linear Equations
Table of Values
x and y intercepts
Slope Intercept Form:
y = mx + b

8. Table of Values x y

9. 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.

10. 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.

11. Table of Values
Example: Given: 2x + 3y = 6
Solve: x y 1 -2

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

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

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

15. 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

16. Table of Values
Plot the points
(0, 2)
(1.5, 1)
(-2, 3 1/3)

17. Table of Values Draw a line through the points
You have graphed the equation:
2x + 3y = 6

18. 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.

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

20. Graph by x and y intercepts
Find the x-intercept by making y = 0.
2x – 3(0) = 6
2x = 6
x = 3
x-intercept = (3, 0)

21. Graph by x and y intercepts
Find the y-intercept by making x = 0.
2(0) + 3y = 6
3y = 6
y = 2
y-intercept = (0, 2)

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

23. Graph by x and y intercepts
Draw a line through the intercepts and you have graphed:
2x + 3y = 6

24. 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.

25. 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.

26. Slope intercept form: y = mx + b
Different Types of slopes
Positive
Negative
Slope of zero
Slope is undefined

27. Slope intercept form: y = mx + b
Examples of: Positive Slopes

28. Slope intercept form: y = mx + b
Examples of: Negative Slopes

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

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

31. Slope intercept form: y = mx + b
The slope formula is used to find the slope given two coordinates.
Slope formula:

32. Slope intercept form: y = mx + b
Example:
Find the slope given:
(2, -3) and (-4, -1)

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

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

35. 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.

36. 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.

37. 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.

38. 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

39. Slope intercept form: y = mx + b
Example:
Graph: 2x +3y = 6 using slope intercept form

40. Slope intercept form: y = mx + b
Step 1: Convert to slope intercept form
2x + 3y = 6
-2x x
3y = -2x + 6
y = (-2/3) x + 2

41. Slope intercept form: y = mx + b
Step 2: Determine your “m” and “b”
y = (-2/3) x + 2
m = (-2/3)
b = 2

42. Slope intercept form: y = mx + b
Step 3:
Plot your
“b” (y-intercept)
b = 2

43. 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

44. 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.

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

46. y= -2x + 3

47. 3y = -4x - 1

48. x – 2y = 4