Download Skill Builder #32

SOLVING LINEAR EQUATIONS
#32
Solving equations involves “undoing” what has been done to create the equation. By
systematically working backward, the value of the variable can be found. Multiplication and
division are inverse (opposite) operations, as are addition and subtraction. For example, to
undo x + 7 = 12, that is, adding 7 to x, subtract 7 from both sides of the equation. The result
is x = 5, which makes x + 7 = 12 true. For 3x = 24, x is multiplied by 3, so divide both
sides by 3 and the result is x = 8. Always remember to follow the correct order of operations.
Example 1
Example 2
4x means 4 times some number x. For
example, suppose 4x = 20. To solve, we
do the inverse of multiplying by 4, which is
dividing by 4. So 20 ÷ 4 = 5, x = 5.
Example 3
4x + 6
4x
x
4(3) + 6
x
5
= 30
To solve, do the inverse of
dividing by 5; that is, multiply
by 5. The result is x = 150.
Example 4
= 18
= 12
=3
= 18
Subtract 6.
Divide by 4.
Substitute 3 for x to
check your answer.
n
2
+ 8 = 14
n
2
=6
Subtract 8.
Multiply by 2.
n = 12
(12)
2
+ 8 = 14
Substitute to check.
Example 5
Solve the equation 2(3x + 3) = 4x + 18.
2(3x + 3)
= 4x + 18
Use the Distributive Property to eliminate the parentheses.
2 · 3x + 2 · 3 = 4x + 18
6x + 6 = 4x + 18
–6
– 6
6x = 4x + 12
– 4x – 4x
2x
2
12
= 2
x =6
Subtract 6 from both sides of the equation.
Subtract 4x from both sides of the equation.
Divide both sides by 2 to get one x.
Substitute the solution into the original equation to check your
answer.
2(3 · 6 + 3) = 4 · 6 + 18
Use the order of operations and simplify.
2(21) = 42
√
Example 6
2
- 3 x – 1 = -3
Solve the equation:
OR
2
2
- 3 x = -2
Add 1 to both sides of the equation.
3
- 3 x = -2
2
3
2
( - 2 )(- 3 ) x = -2(- 2 )
Use reciprocals or division to
solve.
2
2
(- 3 ) x(- 3 ) = -2(- 3 )
x =3
x =3
Problems
Solve these equations. Remember to check your answers.
1.
4x + 3 = 7
2.
2x – 3 = 7
3.
x – 10 = 35
4.
2x – 10 = -4
5.
x
4
+2=6
6.
n
3
7.
8x + 12 = 132
8.
14x – 9 = 75
9.
21 = 4y – 7
10.
16 + 6y = 82
11.
x
3
– 8 = -15
12.
x
15
13.
3(4x – 3) = 63
14.
2(2x + 5) = 33
15.
-3(2x + 4) = -5(x – 3)
16.
-4(-5x – 2) = 16
17.
4x + 2x + 9 = -x + 12
18.
x + 3(x – 1) = x + 18
19.
2x + 4 + x – 7 = 3(x – 7) 20.
4(m – 2) = -3(m – 16)
21.
1
4
24.
- 2 x = -15
22.
25.
2
3
1
3
1
x = 18
23.
-4 x=5
x – 5 = -3
26.
- 3 x – 14 = -10
–6=2
– 7 = -2
x=4
5
2
Solve the remaining equations for the indicated variable.
27.
2y + b = 9 for b
28.
r t = 7 for t
29.
5x – 2d = m for x
30.
3x + 2y = 4 for x
Answers
1.
x=1
2.
x=5
3.
x = 45
4.
x=3
5.
x = 16
6.
n = 24
7.
x = 15
8.
x=6
y = 11
x = -21
12.
x = 75
13.
x=6
y=7
x=
10.
11.
9.
14.
15.
x = -27
17.
x
3
=7
18.
x=7
19.
x = -9
20.
m=8
23.
28.
x = -20
24.
x=6
25.
x=6
29.
x=
30.
x=
2
5
16.
x=
21.
x = 16
22.
x = 27
26.
x = -6
27.
b = 9 – 2y
m + 2d
5
4 – 2y
3