1. 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 __

2. Learn to solve equations using multiplication and division.

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

4. 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 ? 

5. 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 ? 

6. 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 ? 

7. 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 ? 

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

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

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

11. = =  • 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 = 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.

12. = = • 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 = 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.

13. 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.

14. 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

15. 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

16. Lesson Quizzes
Standard Lesson Quiz

17. Lesson Quiz Solve. 1. 3t = 9 2. –15 = 3b t = 3 3. = –7 4. z ÷ 4 = 22
= –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