1. Preview
Warm Up
California Standards
Lesson Presentation

2. Warm Up in Spiral. Solve. 1. x – 16 = 8 2. 7a = 35 3. x = 24
4. y + 21 = 31
x = 24
a = 5 x 12
x = 132
= 11
y = 10

3. AF1.1 Write and solve one-step linear equations in one variable.
Also covered: NS2.1, NS2.2, NS2.4
California
Standards

4. The goal when solving equations that contain fractions is the same as when working with other kinds of numbers—to get the variable by itself on one side of the equation.

5. Additional Example 1A: Solving Equations by Adding or Subtracting
Solve. Write the answer in simplest form. 3 7 5 7
x – =
Think: What is happening to x. How do I undo that?
x – 3 7 = 5 3 7
Since is subtracted from x,
add to both sides.
x – 3 7 + = 5 x = 8 7 1
Simplify.

6. Additional Example 1B: Solving Equations by Adding or Subtracting
Solve. Write the answer in simplest form. 3 8 5 12 +
t = 3 8 + t = 5 12
Since is added to t, 3 8
subtract from both sides. 3 8 + t – = 5 12
t = 10 24 – 9
Find a common denominator.
t = 1 24
Subtract.

7. Additional Example 1C: Solving Equations by Adding or Subtracting
Solve. Write the answer in simplest form. 4 9 1 2
+ r = 4 9 + r 1 2 =
Since is added to r, 4 9
subtract from both sides. 4 9 r 4 9 1 2 4 9 + – = – –
r = 9 18 8
Find a common denominator.
r = 1 18
Subtract.

8. Additional Example 1C Continued
Check. 4 9 1 2
+ r =
Substitute for r. 1 18 4 9 +
1 18 1 2 = ? 8 18 + 1 9 = ?
Find a common denominator. = 9 18 
Add.
You can check that a value is a solution to an equation by substituting the value for the variable. Substitute and solve. Is it correct?
Helpful Hint

9. Solve. Write the answer in simplest form.
Check It Out! Example 1A
Solve. Write the answer in simplest form. 3 8 7 8
x – =
x – 3 8 = 7
Since is subtracted from x, add
to both sides. 3 8
x – 3 8 + = 7
x = 10 8
= 1 1 4
Simplify.

10. + y = Check It Out! Example 1B
Solve. Write the answer in simplest form. 1 4 3 8
+ y = 1 4 + y = 3 8
Since is added to y, subtract
from both sides. 1 4 1 4 3 8 + y – = 3 8 y = – 2
Find a common denominator. y = 1 8
Subtract.

11. Check It Out! Example 1B Continued4 3 8
+ y =
Substitute for y. 1 8 1 4 +
1 8 3 8 = ? 2 8 + 1 3 = ?
Find a common denominator. = 3 8 
Add.

12. Solve. Write the answer in simplest form.
Check It Out! Example 1C
Solve. Write the answer in simplest form. 3 14 2 7 +
t = 3 14 + t = 2 7
Since is added to t, 3 14
subtract from both sides. 3 14 + t – = 2 7
t = – 3 14 4
Find a common denominator. t = 1 14
Subtract.

13. Additional Example 2A: Solving Equations by Multiplying
Solve. Write the answer in simplest form. 3 8 x = 1 4 3 8 = 1 4 x 3 8
Multiply by the reciprocal of . 2 3 8 8 3 1 4 8 3 x =  
Then simplify. 1 2 3 x =
Caution!
To undo multiplying by 3 8
, you can divide by 3 8 8 3
or multiply
by its reciprocal, .

14. Additional Example 2B: Solving Equations by Multiplying
Solve. Write the answer in simplest form. 8 9
4y = 8 9
4y =
Multiply by the reciprocal of 4. 2 1 4 1 4 8 9 = 
Then simplify. 4y  1 2 9
y =

15. Check It Out! Example 2A
Solve. Write the answer in simplest form. 3 4 1 2 x = 3 4 = 1 2 x 3 4
Multiply by the reciprocal of . 2 3 4 . 4 3 1 2 . 4 3 x =
Then simplify. 1 2 3 x =

16. Check It Out! Example 2B
Solve. Write the answer in simplest form. 6 7
3y = 6 7
3y =
Multiply by the reciprocal of 3. 2 y 1 3 6 7 1 3 3 =  
Then simplify. 1 2 7
y =

17. Additional Example 3: Physical Science Application
The amount of copper in brass is of the total weight. If a sample contains 4 ounces of copper, what is the total weight of the sample? 3 4 1 5
Let w represent the total weight of the sample. 3 4
w = 4 1 5
Write an equation. 3 4
w  4 3 1 5 4 3
= 4 
Multiply by the reciprocal of 3 4 7 1 5
Write 4
as an improper
w = 21 5 4 3 
fraction. 1 28 5
w =
or 5 3 5
Then simplify. 3 5
The sample weighs 5
ounces.

18. Check It Out! Example 3
The amount of copper in zinc is of the total weight. If a sample contains 5 ounces of copper, what is the total weight of the sample? 1 4 1 3
Let w represent the total weight of the sample. 1 4
w = 5 3
Write an equation. 1 4
w  4 1 1 3  4 1
= 5
Multiply by the reciprocal of 1 4 1 3
Write 5
as an improper
w = 16 3  4 1
fraction. 64 3
w =
or 21 1 3
Multiply. 1 3
The sample weighs 21
ounces.

19. Solve. Write each answer in simplest form. 1. 2. 3. 4.
Lesson Quiz
Solve. Write each answer in simplest form. 1. 2. 3. 4. 3 8 5 8
x – = 1 7 16 19 32 5 32
y + = x 4 3 7 12 7 5
or 1 = 3 4 1 3 16 9 7
or 1 x = 1
5. Over the course of a week, Marissa ate some apples from a basket on the table. She left 20 apples in the basket. This was five-eighths the number of apples her mother had picked earlier in the week. How many apples did her mother pick? 32