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Chapter 10 Graphing Equations and Inequalities Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 10.5 Equations of Lines Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Slope-Intercept Form When a linear equation in two variables is written in slope-intercept form, y = mx + b slope (0, b), y-intercept then m is the slope of the line and (0, b) is the y-intercept of the line. Note: This form is useful for graphing, since you have a point and the slope readily visible. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 3 Example Use the slope-intercept form to graph the equation 3 y x 2. 5 The slope is 3/5. The y-intercept is –2. Begin by graphing (0, –2), move up 3 units and right 5 units. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 4 Example Find the slope and y-intercept of the line –3x + y = –5. First, we need to solve the linear equation for y. By adding 3x to both sides, y = 3x – 5. Once we have the equation in the form of y = mx + b, we can read the slope and y-intercept. slope is 3 y-intercept is (0, –5) Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 5 Example Find an equation of the line with y-intercept (0, –2) and slope of 1 . We are given the slope and the 3 y-intercept. We let m = 1 and b = –2 and write the 3 equation in slope-intercept form, y = mx + b. y = mx + b 1 y x ( 2) 3 1 y x2 3 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 6 Point-Slope Form The point-slope form of the equation of a line is y y1 m( x x1 ), where m is the slope of the line and (x1, y1) is a point on the line. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 7 Example Find an equation of the line with slope –2 that passes through (–11, –12). Write the equation in slope-intercept form, y = mx + b, and in standard form, Ax + By = C. We substitute the slope and point into the point-slope form of an equation. y – (–12) = – 2(x – (– 11)) Let m = –2 and (x1, y1) = (–11, –12). y + 12 = –2x – 22 y = –2x – 34 Use the distributive property. 2x + y = –34 Add 2x to both sides and we have standard form. Slope-intercept form. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 8 Example Find the equation of the line through (–4, 0) and (6, –1). Write the equation in standard form. First, find the slope. y2 y1 1 0 1 m x2 x1 6 (4) 10 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e continued 9 continued Now substitute the slope and one of the points into the point-slope form of an equation. 1 y 0 ( x (4)) 10 10y = –1(x + 4) Clear fractions by multiplying both sides by 10. 10y = –x – 4 Use the distributive property. x + 10y = –4 Add x to both sides. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 10