The Graph of Ax + By = C

1. Section 3-4
T h e G r a p h o f A x + B y = C

2. Warm-up
Graph the following.
1. y = 3x + 6 2. y = x − 4
2
3

3. Warm-up
Graph the following.
1. y = 3x + 6 2. y = x − 4
2
3

4. Warm-up
Graph the following.
1. y = 3x + 6 2. y = x − 4
2
3

5. Warm-up
Graph the following.
1. y = 3x + 6 2. y = x − 4
2
3
Question: What do you notice about these two graphs?

6. Standard Form of an Equation for a Line:

7. Standard Form of an Equation for a Line: Standard form;
Ax + By = C, where A, B ≠ 0

8. Standard Form of an Equation for a Line: Standard form;
Ax + By = C, where A, B ≠ 0
“Shocking” Theorem:

9. Standard Form of an Equation for a Line: Standard form;
Ax + By = C, where A, B ≠ 0
“Shocking” Theorem: The graph of Ax + By = C,

10. Standard Form of an Equation for a Line: Standard form;
Ax + By = C, where A, B ≠ 0
“Shocking” Theorem: The graph of Ax + By = C, where A, B ≠ 0,

11. Standard Form of an Equation for a Line: Standard form;
Ax + By = C, where A, B ≠ 0
“Shocking” Theorem: The graph of Ax + By = C, where A, B ≠ 0,
is a line!

12. Standard Form of an Equation for a Line: Standard form;
Ax + By = C, where A, B ≠ 0
“Shocking” Theorem: The graph of Ax + By = C, where A, B ≠ 0,
is a line!
Proof:

13. Standard Form of an Equation for a Line: Standard form;
Ax + By = C, where A, B ≠ 0
“Shocking” Theorem: The graph of Ax + By = C, where A, B ≠ 0,
is a line!
Proof: Ax + By = C

14. Standard Form of an Equation for a Line: Standard form;
Ax + By = C, where A, B ≠ 0
“Shocking” Theorem: The graph of Ax + By = C, where A, B ≠ 0,
is a line!
Proof: Ax + By = C
-Ax -Ax

15. Standard Form of an Equation for a Line: Standard form;
Ax + By = C, where A, B ≠ 0
“Shocking” Theorem: The graph of Ax + By = C, where A, B ≠ 0,
is a line!
Proof: Ax + By = C
-Ax -Ax
By = -Ax + C

16. Standard Form of an Equation for a Line: Standard form;
Ax + By = C, where A, B ≠ 0
“Shocking” Theorem: The graph of Ax + By = C, where A, B ≠ 0,
is a line!
Proof: Ax + By = C
-Ax -Ax
By = -Ax + C
B B

17. Standard Form of an Equation for a Line: Standard form;
Ax + By = C, where A, B ≠ 0
“Shocking” Theorem: The graph of Ax + By = C, where A, B ≠ 0,
is a line!
Proof: Ax + By = C
-Ax -Ax
By = -Ax + C
B B
A C
y = − x+
B B

18. Standard Form of an Equation for a Line: Standard form;
Ax + By = C, where A, B ≠ 0
“Shocking” Theorem: The graph of Ax + By = C, where A, B ≠ 0,
is a line!
Proof: Ax + By = C
-Ax -Ax
By = -Ax + C
B B
A C
y = − x+
B B
A
m=−
B

19. Standard Form of an Equation for a Line: Standard form;
Ax + By = C, where A, B ≠ 0
“Shocking” Theorem: The graph of Ax + By = C, where A, B ≠ 0,
is a line!
Proof: Ax + By = C
-Ax -Ax
By = -Ax + C
B B
A C
y = − x+
B B
A C
m=− b=
B B

20. Example 1
State the slope and y-intercept without rewriting the equation
4x + 7y = 21.

21. Example 1
State the slope and y-intercept without rewriting the equation
4x + 7y = 21.
m=

22. Example 1
State the slope and y-intercept without rewriting the equation
4x + 7y = 21.
A
m=−
B

23. Example 1
State the slope and y-intercept without rewriting the equation
4x + 7y = 21.
A 4
m=− =−
B 7

24. Example 1
State the slope and y-intercept without rewriting the equation
4x + 7y = 21.
A 4
m=− =− b=
B 7

25. Example 1
State the slope and y-intercept without rewriting the equation
4x + 7y = 21.
A 4 C
m=− =− b=
B 7 B

26. Example 1
State the slope and y-intercept without rewriting the equation
4x + 7y = 21.
A 4 C 21
m=− =− b= =
B 7 B 7

27. Example 1
State the slope and y-intercept without rewriting the equation
4x + 7y = 21.
A 4 C 21
m=− =− b= = =3
B 7 B 7

28. Example 1
State the slope and y-intercept without rewriting the equation
4x + 7y = 21.
A 4 C 21
m=− =− b= = =3
B 7 B 7
y-int = (0, 3)

29. Some things to be
aware of

30. Some things to be
aware of
*Slope-Intercept:

31. Some things to be
aware of
*Slope-Intercept: We can represent oblique and horizontal lines,
but not vertical

32. Some things to be
aware of
*Slope-Intercept: We can represent oblique and horizontal lines,
but not vertical
*Standard Form:

33. Some things to be
aware of
*Slope-Intercept: We can represent oblique and horizontal lines,
but not vertical
*Standard Form: We can represent all types of lines

34. Some things to be
aware of
*Slope-Intercept: We can represent oblique and horizontal lines,
but not vertical
*Standard Form: We can represent all types of lines
Another benefit of standard form:

35. Some things to be
aware of
*Slope-Intercept: We can represent oblique and horizontal lines,
but not vertical
*Standard Form: We can represent all types of lines
Another benefit of standard form: We can graph by intercepts

36. Example 2
Graph 10x + 6y = 30

37. Example 2
Graph 10x + 6y = 30
x y
0 5
3 0

38. Example 2
Graph 10x + 6y = 30
x y
0 5
3 0

39. Homework

40. Homework
p. 160 #1-26
“The future belongs to those who prepare for it today.” -
Malcolm X