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Solving Equations with Multiplication and Division

This lesson teaches students how to solve one-step equations with multiplication or division. It covers the properties of equality and the use of inverse operations. Examples and practice problems are provided.

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Solving Equations with Multiplication and Division

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  1. Warm Up Lesson Presentation Lesson Quiz

  2. 2 3 Warm Up Evaluate each expression. 1. (–7)(2.8) 2. 0.96 ÷ 6 3. (–9)(–9) 4. 5. 6. –19.6 0.16 81 1 1.8

  3. Sunshine State Standards Prep for MA.912.A.3.2 Identify and apply the…properties of equality. AlsoPrep forMA.912.A.3.1.

  4. Objective Solve one-step equations in one variable by using multiplication or division.

  5. 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. Remember that 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.

  6. j –8 = 3 –24 3 j –8 = Check 3 –8 Additional Example 1A: Solving Equations by Using Multiplication Solve the equation. Since j is divided by 3, multiply both sides by 3 to undo the division. –24 = j To check your solution, substitute –24 for j in the original equation. –8 –8 

  7. n = 2.8 6 n = 2.8 Check 6 16.8 2.8 6 Additional Example 1B: Solving Equations by Using Multiplication Solve the equation. Since n is divided by 6, multiply both sides by 6 to undo the division. n = 16.8 To check your solution, substitute 16.8 for n in the original equation. 2.8 2.8 

  8. p = 10 5 p = 10 Check 5 50 10 5 Check It Out! Example 1a Solve the equation. Check your answer. Since p is divided by 5, multiply both sides by 5 to undo the division. p = 50 To check your solution, substitute 50 for p in the original equation. 10 10 

  9. y –13 = 3 –39 3 y –13 = Check 3 –13 Check It Out! Example 1b Solve the equation. Check your answer. Since y is divided by 3, multiply both sides by 3 to undo the division. –39 = y To check your solution, substitute –39 for y in the original equation. –13 –13 

  10. c = 7 8 c = 7 Check 8 56 7 8 Check It Out! Example 1c Solve the equation. Check your answer. Since c is divided by 8, multiply both sides by 8 to undo the division. c = 56 To check your solution, substitute 56 for c in the original equation. 7 7 

  11. Check 9y = 108 Additional Example 2A: Solving Equations by Using Division Solve the equation. Check your answer. 9y = 108 Since y is multiplied by 9, divide both sides by 9 to undo the multiplication. y = 12 To check your solution, substitute 12 for y in the original equation. 9(12) 108 108 108 

  12. Check –4.8 = –6v Additional Example 2B: Solving Equations by Using Division Solve the equation. Check your answer. –4.8 = –6v Since v is multiplied by –6, divide both sides by –6 to undo the multiplication. 0.8 = v –4.8 –6(0.8) To check your solution, substitute 0.8 for v in the original equation. –4.8 –4.8 

  13. Check 16 = 4c Check It Out! Example 2a Solve the equation. Check your answer. 16 = 4c Since c is multiplied by 4, divide both sides by 4 to undo the multiplication. 4 = c To check your solution, substitute 4 for c in the original equation. 16 4(4) 16 16 

  14. Check 0.5y = –10 Check It Out! Example 2b Solve the equation. Check your answer. 0.5y = –10 Since y is multiplied by 0.5, divide both sides by 0.5 to undo the multiplication. y = –20 To check your solution, substitute –20 for y in the original equation. 0.5(–20) –10 –10 –10 

  15. Check 15k = 75 Check It Out! Example 2c Solve the equation. Check your answer. 15k = 75 Since k is multiplied by 15, divide both sides by 15 to undo the multiplication. k = 5 To check your solution, substitute 5 for k in the original equation. 15(5) 75 75 75 

  16. 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.

  17. 6 5 6 5 6 5 6 5 The reciprocal of is . Since w is multiplied by , multiply both sides by . 5 w = –20 Check 6 –20 Additional Example 3A: Solving Equations That Contain Fractions Solve the equation. 5 w= –20 6 w = –24 To check your solution, substitute –24 for w in the original equation. –20–20 

  18. 3 2 1 3 3 8 1 1 3 16 16 8 8 2 The reciprocal of is 8. Since z is multiplied by , multiply both sides by 8. = z 3 16 1 Check = z To check your solution, substitute for z in the original equation. 8 Additional Example 3B: Solving Equations That Contain Fractions Solve the equation. 3 = z 16 

  19. 1 5 4 4 1 5 1 1 5 5 4 5 The reciprocal of is 5. Since b is multiplied by , multiply both sides by 5. – = b 1 1 = b – Check 4 5 To check your solution, substitute – for b in the original equation. Check It Out! Example 3a Solve the equation. Check your answer. –= b 

  20. 4j 6 Solve the equation. Check your answer. 2 4j is the same as j. = 3 6 6 4 6 4 4 6 4 6 4 6 The reciprocal of is . Since j is multiplied by , multiply both sides by . Check It Out! Example 3b j = 1

  21. Solve the equation. Check your answer. 2 4j = 3 6 4j = Check 6 2 3 Check It Out! Example 3b Continued To check your solution, substitute 1 for j in the original equation. 

  22. 1 1 6 6 The reciprocal of is 6. Since w is multiplied by , multiply both sides by 6 . 1 Check w = 102 6 Check It Out! Example 3c Solve the equation. Check your answer. 1 w= 102 6 w = 612 To check your solution, substitute 612 for w in the original equation. 102102 

  23. 1 Ciro puts of the money he earns from mowing lawns into a college education fund. This year Ciro added $285 to his college education fund. Write and solve an equation to find how much money Ciro earned mowing lawns this year. 4 Additional Example 4: Application one-fourth times earnings equals college fund Write an equation to represent the relationship. Substitute 285 for c. Since m is divided by 4, multiply both sides by 4 to undo the division. Ciro earned $1140 mowing lawns. m = $1140

  24. Check it Out! Example 4 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. A plane began descending 45 miles from the airport. Use the equation to find how high the plane was flying when the descent began. Distance divided by 3 equals height in thousands of feet Write an equation to represent the relationship. Substitute 45 for d. 15 = h The plane was flying at 15,000 ft when the descent began.

  25. Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems

  26. Lesson Quiz: Part I Solve each equation. 1. 2. 3. 8y = 4 4. 126 = –9q 5. 6. 21 2.8 –14 40

  27. 7. A person's weight on Venus is about his or her weight on Earth. Write and solve an equation to find how much a person weighs on Earth if he or she weighs 108 pounds on Venus. 9 10 Lesson Quiz: Part II

  28. Lesson Quiz for Student Response Systems 1. Solve the equation. C. y = 0.45 A. y = 0.0045 D. y = 4.5 B. y = 0.045

  29. Lesson Quiz for Student Response Systems 2. Solve the equation. A. a = –24 C. a = –2 B. a = –10 D. a = 24

  30. Lesson Quiz for Student Response Systems 3. Solve the equation. 6f = 2 A. C. f = 2 B. D. f = 3

  31. Lesson Quiz for Student Response Systems 4. Solve the equation. 150 = –5v A. v = –60 C. v = 30 B. v = –30 D. v = 60

  32. Lesson Quiz for Student Response Systems 5. Solve the equation. A. p = –12 C. p = 6 B. p = –6 D. p = 12

  33. Lesson Quiz for Student Response Systems 6. A boy’s weight on the moon is about his weight on Earth. Identify the equation that shows how much the boy weighs on Earth if he weighs 15 pounds on the moon. A. C. B. D.

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