1. Graphing Ax + By = C
Topic 4.2.2

2. Graphing Ax + By = C 4.2.2 Topic California Standards:
6.0 Students graph a linear equation and compute the x- and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4).
7.0 Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point- slope formula.
What it means for you:
You’ll learn how to graph a straight line by joining two points.
Key Words:
linear equation

3. Graphing Ax + By = C 4.2.2 Topic
Every point on a line is a solution to the equation of the line.
y = 2x
(2, 4)
(–2, –4)
If you know any two solutions (any two coordinate pairs)…
If you know any two solutions (any two coordinate pairs), then you can join the points with a straight line.

4. Graphing Ax + By = C 4.2.2 Topic
Graphing the Line Ax + By = C Using Two Points
The graph of the equation Ax + By = C consists of all points (x, y) whose coordinates satisfy Ax + By = C.
To graph the line, you just need to plot two points on it and join them together with a straight line.

5. Graphing Ax + By = C 4.2.2 Topic Here’s how you go about it:
• Rearrange the equation so it is in the form y = Px + Q.
• Choose two values of x and substitute them into your equation to find the corresponding values of y.
• Plot the two points and draw a straight line through them.
• Plot a third point to check that the line is correct — the point should lie on the line.

6. Graphing Ax + By = C 4.2.2 Topic
Example 1
Plot and label the graph of the equation x – y = –3.
Solution
Rearrange the equation to get y = x + 3.
Choose two values of x, then draw a table to help you find the y-values:
(4, 7)
y = x + 3, so y = = 7 4
(–2, 1)
y = x + 3, so y = –2 + 3 = 1 –2
(x, y) y x
Solution continues…
Solution follows…

7. Graphing Ax + By = C 4.2.2 Topic
Example 1
Plot and label the graph of the equation x – y = –3.
Solution (continued)
When you plot the graph, the line should be straight.
(4, 7)
(–2, 1)
(x, y)
Check your solution:
When x = 1
(1, 4)
y = x + 3 = = 4
So (x, y) = (1, 4)
(1, 4) lies on the line — which means the line is correct.

8. Graphing Ax + By = C 4.2.2 Topic
Example 2
Plot and label the graph of the equation y = –2x – 4.
Solution
Choose two values of x, then draw a table to help you find the y-values:
(2, –8)
y = –2x – 4 = –2(2) – 4 = –8 2
(–2, 0)
y = –2x – 4 = –2(–2) – 4 = 0 –2
(x, y) y x
Solution continues…
Solution follows…

9. Graphing Ax + By = C 4.2.2 Topic
Example 2
Plot and label the graph of the equation y = –2x – 4.
Solution (continued)
Use the points in the table to plot the graph.
(2, –8)
(–2, 0)
(x, y)
Check:
x = 0
y = –2x – 4 = –2(0) – 4
= 0 – 4 = – 4
(0, –4)
(0, –4) lies on the line — which means the line is correct.

10. Graphing Ax + By = C 4.2.2 Topic Guided Practice
Graph the line through the two points in each of Exercises 1–2. 2 1
(–1, –3) and (3, 5)
(–3, 4) and (4, –3)
(–1, –3)
(3, 5)
(–3, 4)
(4, –3)
Solution follows…

11. Graphing Ax + By = C 4.2.2 Topic Guided Practice
Graph and label the lines of the equations in Exercises 3–4.
–x – 2y = 4
2x – 3y = 6 4
2x – 3y = 6
–x – 2y = 4 3
Solution follows…

12. Graphing Ax + By = C 4.2.2 Topic Guided Practice
Graph and label the lines of the equations in Exercises 5–6.
5y – 3x = 15
7y – 2x = –14 5
5y – 3x = 15 6
7y – 2x = –14
Solution follows…

13. Graphing Ax + By = C 4.2.2 Topic Independent Practice
In Exercises 1–4, graph the line through each set of points.
(–1, –2) and (2, 4)
(–1, –1) and (1, 3)
(0, 0) and (2, 6)
(0, –2) and (1, 1) 2 1 3 4
(2, 6)
(2, 4)
(1, 3)
(1, 1)
(0, 0)
(–1, –1)
(–1, –2)
(0, –2)
Solution follows…

14. Graphing Ax + By = C 4.2.2 Topic Independent Practice
Graph and label the lines of the equations in Exercises 5–8.
x + y = 8
y – x = 10
2x + y = –3
5x + y = –12 7
2x + y = –3 6
y – x = 10 8
5x + y = –12 5
x + y = 8
Solution follows…

15. Graphing Ax + By = C 4.2.2 Topic Independent Practice
Graph and label the lines of the equations in Exercises 9–12.
–3x + y = –6
–10x + y = 21
2x – y = –14
6x + 2y = 18 9
–3x + y = –6 10
–10x + y = 21 11
2x – y = –14 12
6x + 2y = 18
Solution follows…

16. Graphing Ax + By = C 4.2.2 Topic Independent Practice
Graph and label the lines of the equations in Exercises 13–16.
8x + 4y = 24
12x – 4y = 8
3x – 9y = –27
2x – 8y = 16 14
12x – 4y = 8 13
8x + 4y = 24 15
3x – 9y = –27 16
2x – 8y = 16
Solution follows…

17. Graphing Ax + By = C 4.2.2 Topic Round Up
It’s easy to make a mistake when working out y-values, so choose x-values that will make the algebra easy (for example, 0 and 1).
And it’s always a good idea to check your line by plotting a third point.