1. The x- and y-Intercepts
Topic 4.2.4

2. The x- and y-Intercepts
Topic
4.2.4
The x- and y-Intercepts
California Standard:
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 a linear inequality (e.g. they sketch the region defined by 2x + 6y < 4).
What it means for you:
You’ll learn about x- and y-intercepts and how to compute them from the equation of a line.
Key Words:
intercept
linear equation

3. The x- and y-Intercepts
Topic
4.2.4
The x- and y-Intercepts –4 –2 2 4 y x
The intercepts of a graph are the points where the graph crosses the axes.
This Topic is all about how to calculate them.

4. The x- and y-Intercepts
Topic
4.2.4
The x- and y-Intercepts
The x-Intercept is Where the Graph Crosses the x-Axis
The x-axis on a graph is the horizontal line through the origin. Every point on it has a y-coordinate of 0. That means that all points on the x-axis are of the form (x, 0).
The x-intercept of the graph of Ax + By = C is the point at which the graph of Ax + By = C crosses the x-axis. –4 –2 2 4
y-axis
x-axis
The x-intercept is here (–1, 0)

5. The x- and y-Intercepts
Topic
4.2.4
The x- and y-Intercepts
Computing the x-Intercept Using “y = 0”
Since you know that the x-intercept has a y-coordinate of 0, you can find the x-coordinate by letting y = 0 in the equation of the line. –4 –2 2 4
y-axis
x-axis
(–1, 0)

6. The x- and y-Intercepts
Topic
4.2.4
The x- and y-Intercepts
Example 1
Find the x-intercept of the line 3x – 4y = 18.
Solution
Let y = 0, then solve for x:
3x – 4y = 18
3x – 4(0) = 18
3x – 0 = 18
3x = 18
x = 6
So (6, 0) is the x-intercept of 3x – 4y = 18.
Solution follows…

7. The x- and y-Intercepts
Topic
4.2.4
The x- and y-Intercepts
Example 2
Find the x-intercept of the line 2x + y = 6.
Solution
Let y = 0, then solve for x:
2x + y = 6
2x + 0 = 6
2x = 6
x = 3
So (3, 0) is the x-intercept of 2x + y = 6.
Solution follows…

8. The x- and y-Intercepts
Topic
4.2.4
The x- and y-Intercepts
Guided Practice
In Exercises 1–8, find the x-intercept.
1. x + y = 5
2. 3x + y = 18
3. 5x – 2y = –10
4. 3x – 8y = –21
5. 4x – 9y = 16
6.15x – 8y = 5
7. 6x – 10y = –8
8. 14x – 6y = 0
x + 0 = 5 Þ x = 5 Þ (5, 0)
3x + 0 = 15 Þ x = 6 Þ (6, 0)
5x – 2(0) = –10 Þ x = –2 Þ (–2, 0)
3x – 8(0) = –21 Þ x = –7 Þ (–7, 0)
4x – 9(0) = 16 Þ x = 4 Þ (4, 0)
15x – 8(0) = 5 Þ x = Þ ( , 0) 1 3
6x – 10(0) = –8 Þ x = – Þ (– , 0) 4 3
14x – 6(0) = 0 Þ x = 0 Þ (0, 0)
Solution follows…

9. The x- and y-Intercepts
Topic
4.2.4
The x- and y-Intercepts
The y-Intercept is Where the Graph Crosses the y-Axis
The y-axis on a graph is the vertical line through the origin. Every point on it has an x-coordinate of 0. That means that all points on the y-axis are of the form (0, y).
The y-intercept of the graph of Ax + By = C is the point at which the graph of Ax + By = C crosses the y-axis. –4 –2 2 4 6
y-axis
x-axis –0
The y-intercept here is (0, 3)

10. The x- and y-Intercepts
Topic
4.2.4
The x- and y-Intercepts
Computing the y-Intercept Using “x = 0”
Since the y-intercept has an x-coordinate of 0, find the y-coordinate by letting x = 0 in the equation of the line. –4 –2 2 4 6
y-axis
x-axis –0
(0, 3)

11. The x- and y-Intercepts
Topic
4.2.4
The x- and y-Intercepts
Example 3
Find the y-intercept of the line –2x – 3y = –9.
Solution
Let x = 0, then solve for y:
–2x – 3y = –9
–2(0) – 3y = –9
0 – 3y = –9
–3y = –9
y = 3
So (0, 3) is the y-intercept of –2x – 3y = –9.
Solution follows…

12. The x- and y-Intercepts
Topic
4.2.4
The x- and y-Intercepts
Example 4
Find the y-intercept of the line 3x + 4y = 24.
Solution
Let x = 0, then solve for y:
3x + 4y = 24
3(0) + 4y = 24
0 + 4y = 24
4y = 24
y = 6
So (0, 6) is the y-intercept of 3x + 4y = 24.
Solution follows…

13. The x- and y-Intercepts
Topic
4.2.4
The x- and y-Intercepts
Guided Practice
In Exercises 9–16, find the y-intercept.
9. 4x – 6y = 24
10. 5x + 8y = 24
11. 8x + 11y = –22
12. 9x + 4y = 48
13. 6x – 7y = –28
14. 10x – 12y = 6
15. 3x + 15y = –3
16. 14x – 5y = 0
4(0) – 6y = 24 Þ y = –4 Þ (0, –4)
5(0) + 8y = 24 Þ y = 3 Þ (0, 3)
8(0) + 11y = –22 Þ y = –2 Þ (0, –2)
9(0) + 4y = 48 Þ y = 12 Þ (0, 12)
6(0) – 7y = –28 Þ y = 4 Þ (0, 4)
10(0) – 12y = 6 Þ y = –0.5 Þ (0, –0.5)
3(0) + 15y = –3 Þ y = –0.2 Þ (0, –0.2)
14(0) – 5y = 0 Þ y = 0 Þ (0, 0)
Solution follows…

14. The x- and y-Intercepts
Topic
4.2.4
The x- and y-Intercepts
Independent Practice
1. Define the x-intercept.
2. Define the y-intercept.
The point at which the graph of a line crosses the y-axis.
The point at which the graph of a line crosses the x-axis.
Solution follows…

15. The x- and y-Intercepts
Topic
4.2.4
The x- and y-Intercepts
Independent Practice
Find the x- and y-intercepts of the following lines:
3. x + y = x – y = 7
5. –x – 2y = x – 3y = 9
7. 3x – 4y = –2x + 3y = 12
9. –5x – 4y = –0.2x + 0.3y = 1
x – 0.2y = – x – y = 6
(9, 0); (0, 9)
(7, 0); (0, –7)
(–4, 0); (0, –2)
(9, 0); (0, –3)
(8, 0); (0, –6)
(–6, 0); (0, 4)
(–5, 0); (0, 3 ) 1 3
(–4, 0); (0, –5) 1 2 2 3
(–12, 0); (0, –9)
(8, 0); (0, –10)
Solution follows…

16. The x- and y-Intercepts
Topic
4.2.4
The x- and y-Intercepts
Independent Practice
13. ( g, 0) is the x-intercept of the line –10x – 3y = 12.
Find the value of g.
14. (0, k) is the y-intercept of the line 2x – 15y = –3.
Find the value of k.
15. The point (–3, b) lies on the line 2y – x = 8. Find the value of b.
16. Find the x-intercept of the line in Exercise 15.
17. Another line has x-intercept (4, 0) and equation 2y + kx = 20. Find the value of k. 3 5 1
g = –2
k = 1
b = 2.5
(–8, 0)
k = 5
Solution follows…

17. The x- and y-Intercepts
Topic
4.2.4
The x- and y-Intercepts
Independent Practice
In Exercises 18-22, use the graph below to help you reach your answer.
18. Find the x- and y-intercepts of line n.
19. Find the x-intercept of line p.
20. Find the y-intercept of line r.
21. Explain why line p does not have a y-intercept.
22. Explain why line r does not have an x-intercept. –6 –4 –2 2 4 6 y x p n r
(1, 0); (0, –4)
(–3, 0)
(0, 2)
Line p is vertical and never crosses the y-axis.
Line r is horizontal and never crosses the x-axis.
Solution follows…

18. The x- and y-Intercepts
Topic
4.2.4
The x- and y-Intercepts
Round Up
Make sure you get the method the right way around — to find the x-intercept, put y = 0 and solve for x, and to find the y-intercept, put x = 0 and solve for y.
In the next Topic you’ll see that the intercepts are really useful when you’re graphing lines from the line equation.