Direct Variation Algebra Lesson 8-9: Solving Equations and Proportions

1. Direct Variation ALGEBRA 1 LESSON 8-9 (For help, go to Lessons 2–7 and 4–1.) Solve each equation for the given variable. 1.nq = m; q2.d = rt; r3.ax + by = 0; y Solve each proportion. 4. = 5. = 6. = 7. = 8. = 9. = 5 8 x 12 4 9 n 45 25 15 y 3 7 n 35 50 8 d 20 36 14 18 63 n 8-9

2. 3.ax + by = 0 ax – ax + by = 0 – ax by = –ax = y = – 1.nq = m2.d = rt = = q = = r r = rt t d t m n nq n d t m n –ax b by b d t ax b 4. = 8x = 5(12) 8x = 60 x = 7.5 5 8 x 12 4 9 n 45 25 15 y 3 5. = 6. = 9n = 4(45) 15y = 25(3) 9n = 180 15y = 75 n = 20 y = 5 Direct Variation ALGEBRA 1 LESSON 8-9 Solutions 8-9

3. 8 d 20 36 7 n 35 50 14 18 63 n 7. = 8. = 35n = 7(50) 20d = 8(36) 35n = 350 20d = 288 n = 10 d = 14.4 9. = 14n = 18(63) 14n = 1134 n = 81 Direct Variation ALGEBRA 1 LESSON 8-9 Solutions (continued) 8-9

4. 1 3 2 3 y = – + xDivide each side by –3. 2 3 y = xDivide each side by –3. The equation has the form y = kx, so the equation is a direct variation. The constant of variation is . 2 3 Direct Variation ALGEBRA 1 LESSON 8-9 Is each equation a direct variation? If it is, find the constant of variation. a. 2x – 3y = 1 –3y = 1 – 2xSubtract 2x from each side. The equation does not have the form y = kx. It is not a direct variation. b. 2x – 3y = 0 –3y = –2xSubtract 2x from each side. 8-9

5. 2 3 – = kDivide each side by –3 to solve for k. 2 3 2 3 y = – xWrite an equation. Substitute – for k in y = kx. 2 3 The equation of the direct variation is y = – x . Direct Variation ALGEBRA 1 LESSON 8-9 Write an equation for the direct variation that includes the point (–3, 2). y = kxUse the general form of a direct variation. 2 = k(–3) Substitute –3 for x and 2 for y. 8-9

6. Relate: The weight varies directly with the mass. When x = 6, y = 59. Define: Let = the mass of an object. Let = the weight of an object. x y Direct Variation ALGEBRA 1 LESSON 8-9 The weight an object exerts on a scale varies directly with the mass of the object. If a bowling ball has a mass of 6 kg, the scale reads 59. Write an equation for the relationship between weight and mass. 8-9

7. y x Write: = kUse the general form of a direct variation. 59 = k(6) Solve for k. Substitute 6 for x and 59 for y. 59 6 = kDivide each side by 6 to solve for k. 59 6 59 6 y = xWrite an equation. Substitute for k in y = kx. 59 6 The equation y = x relates the weight of an object to its mass. Direct Variation ALGEBRA 1 LESSON 8-9 (continued) 8-9

8. xy –1 2 1 2 2 –4 xy –2 1 2 –1 4 –2 y x y x 1 –2 2 –1 = –0.5 = –2 –1 2 2 1 = 2 = –0.5 –4 2 –2 4 = –2 = –0.5 x y No, the ratio is not the same for each pair of data. Direct Variation ALGEBRA 1 LESSON 8-9 For the data in each table, tell whether y varies directly with x. If it does, write an equation for the direct variation. Yes, the constant of variation is –0.5. The equation is y = –0.5x. 8-9

9. Relate: The force of 0.75 lb lifts 48 lb. The force of n lb lifts 210 lb. Define: Let n = the force you need to lift 210 lb. Write: = Use a proportion. Substitute 0.75 for force1, 48 for weight1, and 210 for weight2. 0.75 48 n 210 = 0.75(210) = 48n Use cross products. Solve for n. force2 weight2 force1 weight1 n 3.3 Direct Variation ALGEBRA 1 LESSON 8-9 Suppose a windlass requires 0.75 lb of force to lift an object that weighs 48 lb. How much force would you need to lift 210 lb? You need about 3.3 lb of force to lift 210 lb. 8-9

10. 8 3 xy –1 –2 0 0 1 2 3 –6 xy –1 3 0 0 2 –6 3 –9 b. a. yes; – 4 5 y = x Direct Variation ALGEBRA 1 LESSON 8-9 1. Is each equation a direct variation? If it is, find the constant of variation. a.x + 5y = 10 b. 3y + 8x = 0 no 2. Write an equation of the direct variation that includes the point (–5, –4). 3. For each table, tell whether y varies directly with x. If it does, write an equation for the direct variation. yes; y = –3x no 8-9