1. 3.1 Solving Equations
Algebra I

2. Solve an equation using subtraction
EXAMPLE 1
Solve an equation using subtraction
Solve x + 7 = 4.
x + 7 = 4
Write original equation.
x + 7 – 7 = 4 – 7
Subtract 7 from each side.
x = – 3
Simplify.
CHECK Substitute – 3 for x in the original equation.
x + 7 = 4
Write original equation.
–3 + 7 = 4 ?
Substitute –3 for x.
4 = 4
Simplify. Solution checks.

3. EXAMPLE 2
Solve an equation using addition
Solve x – 12 = 3.
x – 12 =
+12
x = 15

4. Solve an equation using division
EXAMPLE 3
Solve an equation using division
Solve – 6x = 48.
– 6x = 48
Write original equation.
– 6x
– 6 48 =
Divide each side by – 6.
x = – 8
Simplify.

5. GUIDED PRACTICE
for Example 2 and 3
Solve the equation. Check your solution.
1. y + 7 = 10
2. x – 5 = 3
y + 7 = 10
x – 5 = +5
y + 7 – 7 = 10 – 7
y = 3
x =
q – 11 = – 5
q – 11 = – 5
+11
– 11
q = 6

6. GUIDED PRACTICE
for Example 2 and 3
CHECK #1
CHECK #2
x – 5 = 3
y + 7 = 10
8 – 5 = 3 ?
3 + 7 = 10 ?
3 = 3
10 = 10
CHECK #3
q – 11 = –5
6 – 11 = –5 ?
–5 = –5

7. GUIDED PRACTICE
for Example 2 and 3
Solve the equation. Check your solution.
= t – 2.
x = 48.
6 = t – 2
4x = 48 4x 4 48 =
+ 2
x = 12
8 = t
CHECK #4
CHECK #5
4x = 48.
6 = t – 2
6 = 8 – 2 ?
= 48 ?
6 = 6
48 = 48

8. GUIDED PRACTICE
for Example 2 and 3
Solve the equation. Check your solution.
– 65 = – 5y.
w = – 54.
– 65 = – 5y
6w = – 54 6w 6
– 54 =
– 65
– 5
– 5y =
w = – 9
13 = y
CHECK #6
CHECK #7
– 65 = – 5y
6w = – 54
6 – 9 = – 54 ?
– 65 = – 5 ? 13
– 65 = – 65
– 54 = – 54

9. GUIDED PRACTICE
for Example 2 and 3
Solve the equation. Check your solution.
= – 8n.
24 = – 8n 24
– 8
– 8n =
– 3 = n
24 = – 8n
CHECK #8
24 = – 8 ?
– 3
24 = 24

10. Solve an equation using multiplication
EXAMPLE 4
Solve an equation using multiplication
= 5 x 4
Solve
SOLUTION
= 5 x 4
Write original equation. 4 x = 5
Multiply each side by 4.
x = 20
Simplify.

11. Solve the equation. Check your Solution.
GUIDED PRACTICE
for Example 4
Solve the equation. Check your Solution. 9.
= 9 t
– 3
SOLUTION
= 9 t
– 3
Write original equation.
CHECK #9
– 3 t = 9
Multiply each side by –3.
= 9. t
– 3
t = –27
Simplify.
= 9.
– 27
– 3 ?
9 = 9

12. Solve the equation. Check your Solution. 10. 6 c 7 =
GUIDED PRACTICE
for Example 4
Solve the equation. Check your Solution.
10. 6 c 7 =
SOLUTION 6 c 7 =
Write original equation. = 7 c 6
Multiply each side by 7.
CHECK #10 c 7 = 6
42 = c
Simplify. ? = 42 7 6
6 = 6

13. Solve the equation. Check your Solution. 11. 13 z – 2 =
GUIDED PRACTICE
for Example 4
Solve the equation. Check your Solution.
11. 13 z
– 2 =
SOLUTION
– 2 13 = z
Write original equation.
CHECK #11 =
– 2 z 13
Multiply each side by – 2. z
– 2 = 13
– 26 = z
Simplify. ? =
– 26
– 2 13
13 = 13

14. Solve the equation. Check your Solution.
GUIDED PRACTICE
for Example 4
Solve the equation. Check your Solution.
12.
= – 11 a 5
= – 11 a 5
Write original equation.
CHECK #12 5 a =
– 11
Multiply each side by 5.
= – 11 a 5
a = – 55
Simplify.
= – 11
– 55 5 ?
– 11 = – 11

15. Solve an equation by multiplying by a reciprocal
EXAMPLE 5
Solve an equation by multiplying by a reciprocal
x = 4 2 7 –
Solve 7 2 –
The coefficient of x is 7 2 –
The reciprocal of is
x = 4 2 7 –
Write original equation. ( )
x ) = 2 7 – 4
Multiply each side by the 7 2 –
reciprocal,
x = – 14
Simplify.

16. Solve an equation by multiplying by a reciprocal
EXAMPLE 5
Solve an equation by multiplying by a reciprocal
ANSWER
The solution is – 14. Check by substituting – 14 for x in the original equation.
x = 4 2 7 –
CHECK
Write original equation.
(–14) = 4 ? 2 7 –
Substitute –14 for x.
4 = 4
Simplify. Solution checks.

17. Solve the equation. Check your Solution. w = 10 5 6 13. 6 5
GUIDED PRACTICE
for Example 5
Solve the equation. Check your Solution.
w = 10 5 6
13. 6 5
The coefficient of w is 5 6
The reciprocal of is
w = 10 5 6
Write original equation. 5 ( )
w ) = 6 10
CHECK #13 5
Multiply each side by the 6
reciprocal,
w = 10 5 6
w = 12
Simplify.
(12) = 10 ? 5 6
10 = 10

18. Solve the equation. Check your Solution. p = 14 2 3 14. 3 2 .
GUIDED PRACTICE
for Example 5
Solve the equation. Check your Solution.
p = 14 2 3
14. 3 2 .
The coefficient of p is 2 3 is .
The reciprocal of
p = 14 2 3
Write original equation. 2
Multiply each side by the 3
reciprocal, 2 ( )
p ) = 3 14
CHECK #14
p = 14 2 3
p = 21
Simplify.
(21) = 14 ? 2 3
14 = 14

19. Solve the equation. Check your Solution.
GUIDED PRACTICE
for Example 5
Solve the equation. Check your Solution.
15.
9 =
– 3 4 m . 4
– 3
The coefficient of m is 3
– 4
– 3 4
The reciprocal of is .
9 =
– 3 4 m
Write original equation.
CHECK #15
– 3 (
– 4 3 9 )
m ) 4 = 3
Multiply each side by the
– 4
reciprocal,
9 =
– 3 4 m
– 12 = m 4
9 =
– 3
(12) ?
Simplify.
9 = 9

20. Solve the equation. Check your Solution.
GUIDED PRACTICE
for Example 5
Solve the equation. Check your Solution.
16.
– 8 =
– 4 5 v 5
– 4
The coefficient of v is . 4
– 5
– 4 5
The reciprocal of is .
– 8 =
– 4 5 v
Write original equation.
– 4 – (
– 5 4 8)
v ) 5 = 4
Multiply each side by the
– 5
reciprocal,
CHECK #16
– 8 =
– 4 5 v
10 = v
Simplify. 5
8 =
– 4
(10) ? –
–8 = – 8

21. EXAMPLE 6
Write and solve an equation
OLYMPICS
In the 2004 Olympics, Shawn Crawford won the 200 meter dash.
His winning time was seconds. Find his average speed to the nearest tenth of a meter per second.
SOLUTION
Let r represent Crawford's speed in meters per second. Write a verbal model. Then write and solve an equation.

22. EXAMPLE 6
Write and solve an equation
= r
19.79
200 r =
10.1 r
Crawford's average speed was about 10.1 meters per
second.
ANSWER

23. GUIDED PRACTICE for Example 6 17. 100 = 10.1 t 10.1t 100 10.1 = 9.9 t
WHAT IF? In the example 6, suppose Shawn Crawford ran 100 meters at the same average speed he ran the 200 meters. How long would it take him to run 100 meters ? Round your answer to nearest tenth of a second.
Hint: Let t represent Crawford’s time in speed.
= t
Therefore, about 9.9 seconds required Crawford’s to run 100 meters.
10.1t
100
10.1 =
9.9 t