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Algebra I
3.1 Solving Equations
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.
EXAMPLE 2
Solve an equation using addition
Solve x – 12 = 3.
x – 12 =
+12
x
3
+12
= 15
EXAMPLE 3
Solve an equation using division
Solve – 6x = 48.
– 6x = 48
– 6x
48
=
–6
–6
x = –8
Write original equation.
Divide each side by – 6.
Simplify.
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=
y + 7 – 7 = 10 – 7
+5
x
y = 3
3.
q – 11 = – 5
q – 11 = – 5
– 11 +11
q
= 6
3
+5
= 8
GUIDED PRACTICE
for Example 2 and 3
CHECK #1
CHECK #2
y + 7 = 10
x – 5 = 3
? 10
3+7=
? 3
8– 5=
3 = 3
10 = 10
CHECK #3
q – 11 = –5
? –5
6 – 11 =
–5 = –5
GUIDED PRACTICE
for Example 2 and 3
Solve the equation. Check your solution.
4. 6 = t – 2.
5.
6= t–2
+2
4x = 48
4x 48
=
4
4
+2
x = 12
8= t
CHECK #4
6= t–2
? 8–2
6=
6= 6
4x = 48.
CHECK #5
4x = 48.
4
? 48
12 =
48 = 48
GUIDED PRACTICE
for Example 2 and 3
Solve the equation. Check your solution.
6.
– 65 = – 5y.
– 65 = – 5y
– 65
– 5y
=
–5
–5
13 = y
CHECK #6
– 65 = – 5y
7.
6w = – 54.
6w = – 54
6w
– 54
=
6
6
w =–9
CHECK #7
6w = – 54
? – 5 13
– 65 =
6 – 9 =? – 54
– 65 = – 65
– 54 = – 54
GUIDED PRACTICE
for Example 2 and 3
Solve the equation. Check your solution.
8.
24 = – 8n.
24 = – 8n
24
– 8n
=
–8
–8
–3 = n
CHECK #8
24 = – 8n
? –8 –3
24 =
24 = 24
EXAMPLE 4
Solve an equation using multiplication
Solve 4x = 5
SOLUTION
4
x =5
4
x 4
4 = 5
x = 20
Write original equation.
Multiply each side by 4.
Simplify.
GUIDED PRACTICE
for Example 4
Solve the equation. Check your Solution.
9. – 3t = 9
SOLUTION
t =9
–3
Write original equation.
t
–3
– 3 9 Multiply each side by –3.
=
–3
t = –27
Simplify.
CHECK #9
t = 9.
–3
– 27 =? 9.
–3
9 = 9
GUIDED PRACTICE
for Example 4
Solve the equation. Check your Solution.
10. 6 = c
7
SOLUTION
6 = c
7
7 6 = 7
42 = c
Write original equation.
c
7
Multiply each side by 7.
Simplify.
CHECK #10
6 = 7c
6 =? 42
7
6 = 6
GUIDED PRACTICE
for Example 4
Solve the equation. Check your Solution.
11. 13 = z
–2
SOLUTION
z
13 =
–2
– 2 13 = – 2
– 26 = z
Write original equation.
z
–2
Multiply each side by – 2.
Simplify.
CHECK #11
13 = –z2
13 =? – 26
–2
13 = 13
GUIDED PRACTICE
for Example 4
Solve the equation. Check your Solution.
a = – 11
12.
5
a = – 11
5
5
Write original equation.
a
= 5 – 11 Multiply each side by 5.
5
a = – 55
Simplify.
CHECK #12
a = – 11
5
– 55 =? – 11
5
– 11 = – 11
EXAMPLE 5
Solve an equation by multiplying by a reciprocal
2
Solve – 7 x = 4
2
–
x = 4
7
7
7
2
–
(–
x ) =–
( )
2
2 4
7
x = – 14
2
The coefficient of x is –
7
7
2
The reciprocal of – 2 is – 7
Write original equation.
Multiply each side by the
7
reciprocal, –
2
Simplify.
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.
CHECK
2
x = 4
7
?
– 2 (–14) =
4
7
4 = 4
–
Write original equation.
Substitute –14 for x.
Simplify. Solution checks.
GUIDED PRACTICE
for Example 5
Solve the equation. Check your Solution.
5
5
The coefficient of w is
w = 10
13.
6
6
6
5
The reciprocal of 6 is
5
6
(
5
5
w = 10
6
6
5
w)=
( 10)
5
6
w = 12
Write original equation.
Multiply each side by the
reciprocal,
Simplify.
6
5
CHECK #13
5
w = 10
6
5
?
(12) = 10
6
10 = 10
GUIDED PRACTICE
for Example 5
Solve the equation. Check your Solution.
2
2
The coefficient of p is
p = 14
14.
3 .
3
3
2
.
The reciprocal of 3 is
2
2
p = 14
Write original equation.
3
3
(
2
2
p)=
3
p = 21
3
( 14)
2
Multiply each side by the
reciprocal,
Simplify.
3
2
CHECK #14
2
p = 14
3
2
?
(21) = 14
3
14 = 14
GUIDED PRACTICE
for Example 5
Solve the equation. Check your Solution.
–3
–3
15. 9 = 4 m
The coefficient of m is
4 .
–4
–3
.
The reciprocal of 4 is
3
–3
Write original equation.
m
9=
4
–4 –3
–4
( 9 ) = 3 ( 4 m ) Multiply each side by the
3
–4
reciprocal,
– 12 = m
Simplify.
3
CHECK #15
–3
m
9=
4
–3
?
(12)
9 =
4
9 = 9
GUIDED PRACTICE
for Example 5
Solve the equation. Check your Solution.
–4
–4
v
.
The coefficient of v is
16. – 8 = 5
5
–5
–4
.
The reciprocal of 5 is
4
–4
Write original equation.
v
–8=
5
–5 –4
–5
(– 8) = 4 ( 5 v ) Multiply each side by the
CHECK #16
4
–5
reciprocal,
4
–4
v
–8=
5
Simplify.
10 = v
–4
?
(10)
–8 =
5
–8 = – 8
EXAMPLE 6
Write and solve an equation
OLYMPICS
In the 2004 Olympics, Shawn
Crawford won the 200 meter dash.
His winning time was 19.79 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.
EXAMPLE 6
Write and solve an equation
200
=
r
19.79
200 = 19.79 r
19.79 19.79
10.1
r
ANSWER
Crawford's average speed was about 10.1 meters per
second.
GUIDED PRACTICE
17.
for Example 6
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.
100
=
100 = 10.1t
10.1
10.1
t
9.9
10.1
t