Warm Up Write an algebraic expression for each word phrase. 1. A number x decreased by 9 2. 5 times the sum of p and 6 3. 2 plus the product of 8 and n 4. the quotient of 4 and a number c x – 9 5(p + 6)‏ 2 + 8n 4 c __

Learn to solve equations using multiplication and division.

You can solve a multiplication equation using the Division Property of Equality.

Additional Example 1A: Solving Equations Using Division Solve 6x = 48. 6x = 48 Use the Division Property of Equality: Divide both sides by 6. 6x = 48 6 6 1x = 8 1 • x = x x = 8 Check 6x = 48 6(8) = 48 ? Substitute 8 for x. 48 = 48 ? 

Additional Example 1B: Solving Equations Using Division Solve –9y = 45. –9y = 45 –9y = 45 Use the Division Property of Equality: Divide both sides by –9. –9 –9 1y = –5 1 • y = y y = –5 Check –9y = 45 –9(–5) = 45 ? Substitute –5 for y. 45 = 45 ? 

Use the Division Property of Equality: Divide both sides by 9. 9x = 36 Check It Out: Example 1A Solve 9x = 36. 9x = 36 Use the Division Property of Equality: Divide both sides by 9. 9x = 36 9 9 1x = 4 1 • x = x x = 4 Check 9x = 36 9(4) = 36 ? Substitute 4 for x. 36 = 36 ? 

Use the Division Property of Equality: Divide both sides by –3. –3 –3 Check It Out: Example 1B Solve –3y = 36. –3y = 36 –3y = 36 Use the Division Property of Equality: Divide both sides by –3. –3 –3 1y = –12 1 • y = y y = –12 Check –3y = 36 –3(–12) = 36 ? Substitute –12 for y. 36 = 36 ? 

You can solve a division equation using the Multiplication Property of Equality.

Additional Example 2: Solving Equations Using Multiplication b –4 Solve = 5. Use the Multiplication Property of Equality: Multiply both sides by –4. b –4 = 5 –4 • –4 • b = –20 Check b –4 = 5 –20 –4 = 5 ? Substitute –20 for b. 5 = 5 ? 

Check It Out: Example 2 c –3 Solve = 5. Use the Multiplication Property of Equality: Multiply both sides by –3. c –3 = 5 –3 • –3 • c = –15 Check c –3 = 5 –15 –3 = 5 ? Substitute –15 for c. 5 = 5 ? 

= =  • Additional Example 3: Money Application 1 4 x 670 x = 670 1 4 To go on a school trip, Helene has raised $670, which is one-fourth of the amount she needs. What is the total amount needed? fraction of amount raised so far  total amount needed = amount raised so far 1 4 • = x 670 x = 670 1 4 Write the equation. x = 670 1 4 4 • 4 • Multiply both sides by 4. x = 2680 Helene needs $2680 total. Check: The amount raised so far is about $700. She needs about 4 times this amount, or $2800. The answer is reasonable.

= = • Check It Out: Example 3  1 x 750 8 1 x = 750 8 1 8 • x = 750 The school library needs money to complete a new collection. So far, the library has raised $750, which is only one-eighth of what they need. What is the total amount needed? fraction of total amount raised so far total amount needed = amount raised so far  1 8 • = x 750 x = 750 1 8 Write the equation. x = 750 1 8 8 • 8 • Multiply both sides by 8. x = 6000 The library needs $6000 total. Check: The amount raised so far is about $800. She needs about 8 times this amount, or $6400. The answer is reasonable.

Sometimes it is necessary to solve equations by using two inverse operations. For instance, the equation 6x  2 = 10 has multiplication and subtraction. Variable term 6x  2 = 10 Multiplication Subtraction To solve this equation, add to isolate the term with the variable in it. Then divide to solve.

Additional Example 4: Solving a Two-Step Equation Solve 3x + 2 = 14. Step 1: 3x + 2 = 14 Subtract 2 to both sides to isolate the term with x in it. – 2 – 2 3x = 12 Step 2: 3x = 12 Divide both sides by 3. 3 3 x = 4

Check It Out: Example 4 Solve 4y + 5 = 29. Step 1: 4y + 5 = 29 Subtract 5 from both sides to isolate the term with y in it. – 5 – 5 4y = 24 Step 2: 4y = 24 Divide both sides by 4. 4 4 y = 6

Lesson Quizzes Standard Lesson Quiz

Lesson Quiz Solve. 1. 3t = 9 2. –15 = 3b t = 3 3. = –7 4. z ÷ 4 = 22 3. = –7 4. z ÷ 4 = 22 5. A roller coaster descends a hill at a rate of 80 feet per second. The bottom of the hill is 400 feet from the top. How long will it take the coaster rides to reach the bottom? t = 3 b = –5 x –4 x = 28 z = 88 5 seconds