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**The x- and y-Intercepts**

Topic 4.2.4

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**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

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**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.

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**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)

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**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)

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**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…

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**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…

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**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…

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**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)

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**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)

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**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…

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**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…

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**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…

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**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…

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**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…

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**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…

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**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…

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**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.

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