The Slope-Intercept Form of a Line Topic 4.4.3

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line California Standard: 8.0 Students understand the concepts of parallel lines and perpendicular lines and how their slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point. What it means for you: You’ll plot graphs and solve equations using a method called the slope-intercept form of an equation. Key Words: slope-intercept form intercept

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line Now that you’ve practiced calculating the slope and intercept of a line, you can use the two things together to make plotting graphs easier. –6 –4 –2 2 4 6 y x

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line The Slope-Intercept Form of a Line: y = mx + b In the slope-intercept form the y is alone on one side of the equation. The slope-intercept form of the equation is: y = mx + b Here are a few examples of equations in the slope-intercept form: y = 3x + 2 y = – x + 1 y = –x + 1 y = x – 4 2 3 1

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line Guided Practice In Exercises 1–6, decide whether each equation is in slope-intercept form or not. 1. y = 3x + 7 2. 3x + 4y = 7 3. y – 3 = 2(x – 4) 4. y – 8 = 3(x – 4) 5. y = x + 18 6. y = –4x – 1 yes no no no 3 2 yes yes Solution follows…

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line The Slope-Intercept Form Makes It Easy to Plot Graphs The slope-intercept form of an equation is really useful for plotting graphs because m is the slope of the line and b is the y-coordinate of the y-intercept. y = mx + b y-coordinate of y-intercept slope

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line Example 1 1 2 1 2 –6 –4 –2 2 4 6 y x y = x + 2 1 Plot the graph of y = x + 2. Solution y = mx + b = x + 2 1 2 — for a slope of , go up 1 unit for every 2 units across. Slope = m = 1 2 y-coordinate of the y-intercept = b = 2 — so the y-intercept is (0, 2). Solution follows…

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line Example 2 –6 –4 –2 2 4 6 y x y = –3x – 4 Plot the graph of y = –3x – 4. Solution y = mx + b = –3x – 4 Slope = m = –3 — for a slope of –3, go down 3 units for every unit across. y-coordinate of the y-intercept = b = –4 — so the y-intercept is (0, –4). Solution follows…

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line Guided Practice 7. In the equation y = 8x + 5, find the slope. 8. In the equation y = –x + 10, find the slope. 9. In the equation y = 2x + 5, find the y-intercept. 10. In the equation y = 7b – 3, find the y-intercept. m = 8 m = –1 y-intercept = (0, 5) y-intercept = (0, –3) Solution follows…

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line Guided Practice In Exercises 11–14, plot each equation on a graph. 11. y = 2x + 3 12. y = x – 6 13. y = –7x – 8 14. y = – x – 4 –6 –4 –2 2 4 6 y x –8 –6 –4 –2 2 4 6 y x 13 12 11 14 1 3 Solution follows…

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line Guided Practice In Exercises 15–18, plot each equation on a graph. 15. y = – x + 6 16. y = x – 3 17. y = x 18. y = 6 1 2 –6 –4 –2 2 4 6 y x 17 16 15 18 1 5 3 4 Solution follows…

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line Solve for y to Get the Slope-Intercept Form OK, so you know the slope-intercept form of an equation makes drawing graphs a lot easier. The trouble is, you’ll often be given an equation which isn’t in slope-intercept form. To get the equation into slope-intercept form, solve for y.

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line Example 3 A line has the equation Ax + By = C, where B ¹ 0. Solve this equation for y, justifying each step. Solution Ax + By = C Given equation Ax – Ax + By = –Ax + C Subtraction property of equality By = –Ax + C Division property of equality Solution continues… Solution follows…

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line Example 3 A line has the equation Ax + By = C, where B ¹ 0. Solve this equation for y, justifying each step. Solution (continued) The equation is now in slope-intercept form. The slope, m = – and the y-coordinate of the y-intercept, b = – . A B C

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line Example 4 Determine the slope and y-intercept of the line 2x – 3y = 9. Solution Step 1: Solve the given equation for y. 2x – 3y = 9 –3y = –2x + 9 y = y = x – 3 2 3 Now you’ve got the equation in slope-intercept form, y = mx + b. Solution continues… Solution follows…

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line Example 4 Determine the slope and y-intercept of the line 2x – 3y = 9. Solution (continued) y = x – 3 2 3 Step 1: Step 2: Get the slope and y-intercept from the equation. The slope, m = . 2 3 The y-coordinate of the y-intercept, b = –3. So, the y-intercept = (0, –3).

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line Guided Practice In Exercises 19-24, find the slope and y-intercept of the line. 19. 3x + 3y = 9 20. 2y – 6x = 10 21. 2x – 2y = 5 22. –7y + 5x = 14 23. –2y + 3x = 8 24. –5x + 4y = 12 m = –1, (0, 3) x + y = 3 Þ y = –x + 3 m = 3, (0, 5) y – 3x = 5 Þ y = 3x + 5 m = 1, (0, – ) 5 2 x – y = Þ y = x – m = , (0, –2) 5 7 y – x = –2 Þ y = x – 2 m = , (0, –4) 3 2 y – x = –4 Þ y = x – 4 m = , (0, 3) 5 4 – x + y = 3 Þ y = x + 3 Solution follows…

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line Guided Practice Write down the equations of the following lines in slope-intercept form. 25. The line with slope 4 that passes through the point (0, 2). 26. The line with slope 2 that passes through the point (0, –6). 27. The line with slope –3 that passes through the point (0, 1). y = 4x + 2 y = 2x – 6 y = –3x + 1 6 7 28. The line with slope – that passes through the point (0, –3). 6 7 y = – x – 3 Solution follows…

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line Independent Practice In Exercises 1–10, find the slope and y-intercept of each equation that’s given. 1. y = x – 5 2. y = x + 1 1 2 2 3 m = , (0, –5) 1 2 m = , (0, 1) 2 3 3. y = 3x + 6 4. –6y = 3x + 12 5. y = 2(x + 2) 6. y – 4 = 2(x + 1) 7. 7 – y = 5(x + 4) 8. y – 3 = 2(x – 9) 9. 3x + 4y = 8 10. 2x + 3y = 9 m = – , (0, –2) 1 2 m = 3, (0, 6) m = 2, (0, 4) m = 2, (0, 6) m = –5, (0, –13) m = 2, (0, –15) 3 4 m = – , (0, 2) m = – , (0, 3) 2 3 Solution follows…

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line Independent Practice In Exercises 11–15, plot the graph of the given equation. 11. y = x + 5 12. y = – x – 6 13. y = x + 2 14. y = –x + 2 15. y = 2x 1 3 –6 –4 –2 2 4 6 y x 15 13 11 14 12 1 3 Solution follows…

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line Independent Practice In Exercises 16–20, write the equations of the lines in slope-intercept form. 16. A line with slope that passes through the point (0, 4). 17. A line with slope that passes through the point (0, –2). 4 3 y = x + 4 4 3 1 2 y = x – 2 1 2 18. 4x + 2y = 8 19. 6x – 3y = 15 20. 3x – 4y = –16 y = –2x + 4 y = 2x – 5 y = x + 4 3 4 Solution follows…

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line Independent Practice In Exercises 21–22, write the equations of the lines in slope-intercept form. 21. The line with slope 0 that passes through the point (2, 6) 22. The line with slope 2 that passes through the point (6, 3) y = 6 y = 2x – 9 Solution follows…

The Slope-Intercept Form of a Line Topic 4.4.3 The Slope-Intercept Form of a Line Round Up If m = 0, the slope-intercept form of the equation becomes y = b — the equation of a horizontal line. If b = 0, the equation becomes y = mx, meaning the line passes through the origin.