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## Solving Equations by Multiplying or Dividing

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**Solving Equations byMultiplying or Dividing**CONFIDENTIAL**Warm Up**Simplify: 1) a = -2.8 1) -5.2 + a = -8 2) 90 = -22 + a 2) a = 112 3) -12 + f = 3 3) f = -9 4) -1200 + j = 345 4) j = 1545 CONFIDENTIAL**Introduction**Solving an equation that contains multiplication or division is similar to solving an equation that contains addition or subtraction. Use inverse operations to undo the operations on the variable. An equation is like a balanced scale. To keep the balance, whatever you do on one side of the equation, you must also do on the other side. CONFIDENTIAL**-4 = k**-5 -4 20 -5 -4 -4 Solving Equations by Using Multiplication Solve each equation. Check your answer. A) -4 = k -5 (-5) (-4) = (-5) ( k) -5 Since k is divided by -5, multiply both sides by -5 to undo the division. 20 = k Check To check your solution, substitute 20 for k in the original equation. CONFIDENTIAL**m= 1.5**3 4.5 3 1.5 1.51.5 B) m= 1.5 3 (3) × m= (3) × 1.5 3 Since m is divided by 3, multiply both sides by 3 to undo the division. m = 4.5 Check To check your solution, substitute 1.5 for m in the original equation. CONFIDENTIAL**Now you try!**Solve each equation. Check your answer. 1) p= 10 5 1) p = 50 2) y = -39 2) -13 = y 3 3) c = 56 3) c= 7 8 CONFIDENTIAL**7x = 56**7(8) 56 56 56 Solving Equations by Using Division Solve each equation. Check your answers. A) 7x = 56 7x = 56 Since x is multiplied by 7, divide both sides by 7 to undo the multiplication. 7x = 56 7 7 x = 8 Check To check your solution, substitute 8 for x in the original equation. CONFIDENTIAL**13 = -2w**13 -2(6.5) 1313 B) 13 = -2w 13 = -2w Since w is multiplied by -2, divide both sides by -2 to undo the multiplication. 13 = -2w -2 -2 -6.5 = w Check To check your solution, substitute 8 for x in the original equation. CONFIDENTIAL**Now you try!**Solve each equation. Check your answer. 1) 16 = 4c 1) c = 4 2) 0.5y = -10 2) y = -20 3) 15k = 75 3) k = 5 CONFIDENTIAL**Remember that dividing is the same as multiplying by the**reciprocal. When solving equations, you will sometimes find it easier to multiply by a reciprocal instead of dividing. This is often true when an equation contains fractions. CONFIDENTIAL**Solving Equations That Contain Fractions**Solve each equation. A) 5v= 35 9 9 × 5v = 9 ×35 5 9 5 The reciprocal of 5/9 is 9/5 . Since v is multiplied by 5/9 , multiply both sides by 9/5. v = 63 CONFIDENTIAL**B) 5= 4y**2 3 5= 4y 2 3 The reciprocal of 4/3 is 3/4 . Since v is multiplied by 4/3 , multiply both sides by 3/4. 3 × 5 = 3 ×4y 4 2 4 3 15 = y 8 CONFIDENTIAL**Now you try!**Solve each equation. Check your answer. 1) -1 = b 4 5 1) b = -1.25 2) 4j = 2 6 3 2) j = 1 3) w = 612 3) w = 102 6 CONFIDENTIAL**(3)d = (3) × 10**3 The distance in miles from the airport that a plane should begin descending, divided by 3, equals the plane’s height above the ground in thousands of feet. If a plane is 10,000 feet above the ground, write and solve an equation to find the distance at which the pilot should begin descending. Solution: Distance divided by 3 equals height in thousands of feet. d= h 3 Write an equation to represent the relationship. d= 10 3 Substitute 10 for h. Since d is divided by 3, multiply both sides by 3 to undo the division. d = 30 The pilot should begin descending 30 miles from the airport. CONFIDENTIAL**Properties of Equality**4 4 c c CONFIDENTIAL**Now you try!**An orange contains about 80 milligrams of vitamin C, which is 10 times as much as an apple contains. Write and solve an equation to find the amount of vitamin C in an apple. 10x = 80 x = 10 milligrams CONFIDENTIAL**Assignments**Solve each equation. Check your answer: 1) K = 8 4 1) k = 32 1) k = 32 2) z = -9 3 2) z = -27 3) -2 = w -7 3) w = 14 4) 6 = t -5 4) t = - 30 CONFIDENTIAL**Factor the trinomial. Check your answer.**5) g = 10 1.9 5) g = 19 6) 2.4 = b 5 6) b = 12 7) 4x = 28 7) x = 7 8) -64 = 8c 8) c = - 8 9) j = 5 9) -9 j = -45 CONFIDENTIAL**10) The Baseball Birthday Batter Package at a minor league**ballpark costs $192. The package includes tickets, drinks, and cake for a group of 16 children. Write and solve an equation to find the cost per child. 10) 16 c =$192 c = $12 CONFIDENTIAL**Let’s review**Solving an equation that contains multiplication or division is similar to solving an equation that contains addition or subtraction. Use inverse operations to undo the operations on the variable. An equation is like a balanced scale. To keep the balance, whatever you do on one side of the equation, you must also do on the other side. CONFIDENTIAL**-4 = k**-5 -4 20 -5 -4 -4 review Solving Equations by Using Multiplication Solve each equation. Check your answer. A) -4 = k -5 (-5) (-4) = (-5) ( k) -5 Since k is divided by -5, multiply both sides by -5 to undo the division. 20 = k Check To check your solution, substitute 20 for k in the original equation. CONFIDENTIAL**m= 1.5**3 4.5 3 1.5 1.51.5 review B) m= 1.5 3 (3) × m= (3) × 1.5 3 Since m is divided by 3, multiply both sides by 3 to undo the division. m = 4.5 Check To check your solution, substitute 1.5 for m in the original equation. CONFIDENTIAL**7x = 56**7(8) 56 56 56 review Solving Equations by Using Division Solve each equation. Check your answers. A) 7x = 56 7x = 56 Since x is multiplied by 7, divide both sides by 7 to undo the multiplication. 7x = 56 7 7 x = 8 Check To check your solution, substitute 8 for x in the original equation. CONFIDENTIAL**13 = -2w**13 -2(6.5) 1313 review B) 13 = -2w 13 = -2w Since w is multiplied by -2, divide both sides by -2 to undo the multiplication. 13 = -2w -2 -2 -6.5 = w Check To check your solution, substitute 8 for x in the original equation. CONFIDENTIAL**review**Remember that dividing is the same as multiplying by the reciprocal. When solving equations, you will sometimes find it easier to multiply by a reciprocal instead of dividing. This is often true when an equation contains fractions. CONFIDENTIAL**review**Solving Equations That Contain Fractions Solve each equation. A) 5v= 35 9 9 × 5v = 9 ×35 5 9 5 The reciprocal of 5/9 is 9/5 . Since v is multiplied by 5/9 , multiply both sides by 9/5. v = 63 CONFIDENTIAL**review**B) 5= 4y 2 3 5= 4y 2 3 The reciprocal of 4/3 is 3/4 . Since v is multiplied by 4/3 , multiply both sides by 3/4. 3 × 5 = 3 ×4y 4 2 4 3 15 = y 8 CONFIDENTIAL**(3)d = (3) × 10**3 The distance in miles from the airport that a plane should begin descending, divided by 3, equals the plane’s height above the ground in thousands of feet. If a plane is 10,000 feet above the ground, write and solve an equation to find the distance at which the pilot should begin descending. Solution: Distance divided by 3 equals height in thousands of feet. d= h 3 Write an equation to represent the relationship. d= 10 3 Substitute 10 for h. Since d is divided by 3, multiply both sides by 3 to undo the division. d = 30 The pilot should begin descending 30 miles from the airport. CONFIDENTIAL**review**Properties of Equality 4 4 c c CONFIDENTIAL**You did a great job today!**CONFIDENTIAL