Solving Equations Clearing Frations and Decimals

2.3. SOLVING EQUATIONS: CLEARING FRATIONS AND DECIMALS73
2.3
Solving Equations: Clearing Frations and Decimals
1. Multiply 16 and 9 to get 144,
9
9
x = 16 ·
16
x
2
2
144
=
x
2
= 72x
then divide 144 by 2 to get 72.
Associative property of multiplication.
Multiply: 16 · 9 = 144.
Divide: 144/2 = 72.
Alternate solution: Divide 2 into 16 to get 8, then multiply 8 by 9 to get
72.
16
9
9
x = 16 ·
x
2
2
= (8 · 9) x
= 72x
Associative property of multiplication.
Divide: 16/2 = 8.
Multiply: 8 · 9 = 72.
Note that the second method is more efficient, because it involves smaller
numbers, making it easier to perform the steps mentally. That is, write
9
x = 72x,
16
2
without writing down any steps.
3. Multiply 14 and 3 to get 42, then divide 42 by 2 to get 21.
3
3
14
x = 14 ·
x
Associative property of multiplication.
2
2
42
=
x
Multiply: 14 · 3 = 42.
2
= 21x
Divide: 42/2 = 21.
Alternate solution: Divide 2 into 14 to get 7, then multiply 7 by 3 to get
21.
14
3
3
x = 14 ·
x
2
2
= (7 · 3) x
= 21x
Associative property of multiplication.
Divide: 14/2 = 7.
Multiply: 7 · 3 = 21.
Note that the second method is more efficient, because it involves smaller
numbers, making it easier to perform the steps mentally. That is, write
3
x = 21x,
14
2
without writing down any steps.
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CHAPTER 2. SOLVING LINEAR EQUATIONS
74
5. Multiply 70 and 9 to get 630, then divide 630 by 7 to get 90.
9
9
70
x = 70 ·
x
Associative property of multiplication.
7
7
630
x
Multiply: 70 · 9 = 630.
=
7
= 90x
Divide: 630/7 = 90.
Alternate solution: Divide 7 into 70 to get 10, then multiply 10 by 9 to
get 90.
9
9
x = 70 ·
70
x
Associative property of multiplication.
7
7
= (10 · 9) x
= 90x
Divide: 70/7 = 10.
Multiply: 10 · 9 = 90.
Note that the second method is more efficient, because it involves smaller
numbers, making it easier to perform the steps mentally. That is, write
9
x = 90x,
70
7
without writing down any steps.
7. Clear fractions from the equation by multiplying both sides by the least
common denominator. The least common denominator in this case is 21.
9
1
5
− x− =
Original equation.
7
3
3
9
1
5
21 − x −
= 21
Multiply both sides by 21.
7
3
3
1
5
9
21 − x − 21
= 21
Distribute the 21 on each side.
7
3
3
−27x − 7 = 35
Multiply.
Note that the fractions are now cleared from the equation. Next, isolate all
terms containing the variable x on one side of the equation. We can remove
the term −7 from the left-hand side by adding 7 to both sides of the equation.
−27x − 7 + 7 = 35 + 7
−27x = 42
Add 7 to both sides.
Simplify both sides.
Finally, to “undo” multiplying by −27, divide both sides of the equation by
−27.
42
−27x
=
−27
−27
14
x=−
9
Second Edition: 2012-2013
Divide both sides by −27.
Simplify both sides.
2.3. SOLVING EQUATIONS: CLEARING FRATIONS AND DECIMALS75
9. Clear fractions from the equation by multiplying both sides by the least
common denominator. The least common denominator in this case is 9.
7
5
2
4
x+ = x−
9
3 3
3
7
5
2
4
9
x+
x−
=9
3
9
3
3
4
5
2
7
x +9
x −9 −
=9
9
3
9
3
3
21x + 5 = 6x − 12
Original equation.
Multiply both sides by 9.
Distribute the 9 on each side.
Multiply.
Note that the fractions are now cleared from the equation. Next, isolate all
terms containing the variable x on one side of the equation. We can remove
the term 6x from the right-hand side by subtracting 6x from both sides of the
equation.
21x + 5 − 6x = 6x − 12 − 6x
15x + 5 = −12
Subtract 6x from both sides.
Simplify both sides.
Next, we can remove the term 5 from the left-hand side by subtracting 5 from
both sides of the equation.
15x + 5 − 5 = −12 − 5
15x = −17
Subtract 5 from both sides.
Simplify both sides.
Finally, to “undo” multiplying by 15, divide both sides of the equation by 15.
15x
−17
=
15
15
17
x=−
15
Divide both sides by 15.
Simplify both sides.
11. Clear fractions from the equation by multiplying both sides by the least
common denominator. The least common denominator in this case is 28.
9
8
3
x− =
4
7
2
9
8
3
x−
28
= 28
4
7
2
8
3
9
x − 28
= 28
28
4
7
2
63x − 32 = 42
Original equation.
Multiply both sides by 28.
Distribute the 28 on each side.
Multiply.
Note that the fractions are now cleared from the equation. Next, isolate all
terms containing the variable x on one side of the equation. We can remove the
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CHAPTER 2. SOLVING LINEAR EQUATIONS
76
term −32 from the left-hand side by adding 32 to both sides of the equation.
63x − 32 + 32 = 42 + 32
63x = 74
Add 32 to both sides.
Simplify both sides.
Finally, to “undo” multiplying by 63, divide both sides of the equation by 63.
63x
74
=
63
63
74
x=
63
Divide both sides by 63.
Simplify both sides.
13. Clear fractions from the equation by multiplying both sides by the least
common denominator. The least common denominator in this case is 12.
3
8
− x=−
4
3
3
8
12 − x = 12 −
4
3
−9x = −32
Original equation.
Multiply both sides by 12.
Multiply.
To “undo” multiplying by −9, divide both sides of the equation by −9.
−32
−9x
=
−9
−9
32
x=
9
Divide both sides by −9.
Simplify.
15. Clear fractions from the equation by multiplying both sides by the least
common denominator. The least common denominator in this case is 20.
3
6
=
4
5 3
6
20 x +
= 20
4
5
6
3
= 20
20x + 20
4
5
x+
20x + 15 = 24
Original equation.
Multiply both sides by 20.
On the left, distribute the 20.
Multiply.
Next, isolate all terms containing x on one side of the equation.
20x + 15 − 15 = 24 − 15
20x = 9
Second Edition: 2012-2013
Subtract 15 from both sides.
Simplify both sides.
2.3. SOLVING EQUATIONS: CLEARING FRATIONS AND DECIMALS77
Finally, to “undo” multiplying by 20, divide both sides of the equation by 20.
20x
9
=
20
20
9
x=
20
Divide both sides by 20.
Simplify both sides.
17. Clear fractions from the equation by multiplying both sides by the least
common denominator. The least common denominator in this case is 60.
4
3
8
1
Original equation.
− x− =− x−
3
3
4
5
1
4
8
3
60 − x −
= 60 − x −
Multiply both sides by 60.
3
3
4
5
4
8
1
3
60 − x − 60
= 60 − x − 60 −
Distribute the 60 on each side.
3
3
4
5
−20x − 80 = −45x − 96
Multiply.
Note that the fractions are now cleared from the equation. Next, isolate all
terms containing the variable x on one side of the equation. We can remove
the term −45x from the right-hand side by adding 45x to both sides of the
equation.
−20x − 80 + 45x = −45x − 96 + 45x
25x − 80 = −96
Add 45x to both sides.
Simplify both sides.
Next, we can remove the term −80 from the left-hand side by adding 80 to
both sides of the equation.
25x − 80 + 80 = −96 + 80
25x = −16
Add 80 to both sides.
Simplify both sides.
Finally, to “undo” multiplying by 25, divide both sides of the equation by 25.
25x
−16
=
25
25
16
x=−
25
Divide both sides by 25.
Simplify both sides.
19. At a minimum, we need to move each decimal point two places to the right
in order to clear the decimals from the equation. Consequently, we multiply
both sides of the equation by 100.
2.39x + 0.71 = −1.98x + 2.29
100(2.39x + 0.71) = 100(−1.98x + 2.29)
239x + 71 = −198x + 229
Original Equation.
Multiply both sides by 100.
Distribute the 100.
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CHAPTER 2. SOLVING LINEAR EQUATIONS
78
Note that the decimals are now cleared from the equation. Next, isolate all
terms containing the variable x on one side of the equation. Remove the term
−198x from the right-hand side by adding 198x to both sides of the equation.
239x + 71 + 198x = −198x + 229 + 198x
437x + 71 = 229
Add 198x to both sides.
Simplify both sides.
Subtract 71 from to eliminate the term 71 from the left-hand side of the equation.
437x + 71 − 71 = 229 − 71
437x = 158
Subtract 71 from both sides.
Simplify both sides.
Finally, to “undo” multiplying by 437, divide both sides of the equation by
437.
437x
158
=
437
437
158
x=
437
Divide both sides by 437.
Simplify.
21. At a minimum, we need to move each decimal point two places to the right
in order to clear the decimals from the equation. Consequently, we multiply
both sides of the equation by 100.
0.4x − 1.55 = 2.14
100(0.4x − 1.55) = 100(2.14)
40x − 155 = 214
Original Equation.
Multiply both sides by 100.
Distribute the 100.
Note that the decimals are now cleared from the equation. We can continue
by adding 155 to both sides of the equation.
40x − 155 + 155 = 214 + 155
40x = 369
369
40x
=
40
40
369
x=
40
Add 155 to both sides.
Simplify both sides.
Divide both sides by 40.
Simplify.
23. At a minimum, we need to move each decimal point two places to the right
in order to clear the decimals from the equation. Consequently, we multiply
both sides of the equation by 100.
2.6x − 2.54 = −2.14x
100(2.6x − 2.54) = 100(−2.14x)
260x − 254 = −214x
Second Edition: 2012-2013
Original Equation.
Multiply both sides by 100.
Distribute the 100.
2.3. SOLVING EQUATIONS: CLEARING FRATIONS AND DECIMALS79
Note that the decimals are now cleared from the equation. Next, isolate all
terms containing the variable x on one side of the equation. Remove the term
−214x from the right-hand side of the equation by adding 214x to both sides
of the equation.
260x − 254 + 214x = −214x + 214x
474x − 254 = 0
Add 214x to both sides.
Simplify both sides.
Add 254 to both sides to remove the term −254 from the left-hand side of the
equation.
474x − 254 + 254 = 0 + 254
474x = 254
Add 254 to both sides.
Simplify both sides.
Finally, to “undo” multiplying by 474, divide both sides of the equation by
474.
474x
254
=
474
474
127
x=
237
Divide both sides by 474.
Simplify.
25. At a minimum, we need to move each decimal point one place to the right
in order to clear the decimals from the equation. Consequently, we multiply
both sides of the equation by 10.
0.7x = −2.3x − 2.8
10(0.7x) = 10(−2.3x − 2.8)
7x = −23x − 28
Original Equation.
Multiply both sides by 10.
Distribute the 10.
Note that the decimals are now cleared from the equation. Next, isolate all
terms containing the variable x on one side of the equation. We can remove
the term −23x from the right-hand side by adding 23x to both sides of the
equation.
7x + 23x = −23x − 28 + 23x
30x = −28
Add 23x to both sides.
Simplify both sides.
To “undo” multiplying by 30, divide both sides of the equation by 30.
30x
−28
=
30
30
14
x=−
15
Divide both sides by 30.
Simplify.
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CHAPTER 2. SOLVING LINEAR EQUATIONS
80
27. At a minimum, we need to move each decimal point one place to the right
in order to clear the decimals from the equation. Consequently, we multiply
both sides of the equation by 10.
−4.8x − 2.7 = −1.9
10(−4.8x − 2.7) = 10(−1.9)
−48x − 27 = −19
Original Equation.
Multiply both sides by 10.
Distribute the 10.
Note that the decimals are now cleared from the equation. We can continue
by adding 27 to both sides of the equation.
−48x − 27 + 27 = −19 + 27
−48x = 8
−48x
8
=
−48
−48
1
x=−
6
Add 27 to both sides.
Simplify both sides.
Divide both sides by −48.
Simplify.
29. At a minimum, we need to move each decimal point one place to the right
in order to clear the decimals from the equation. Consequently, we multiply
both sides of the equation by 10.
1.7x + 2.1 = −1.6x + 2.5
10(1.7x + 2.1) = 10(−1.6x + 2.5)
17x + 21 = −16x + 25
Original Equation.
Multiply both sides by 10.
Distribute the 10.
Note that the decimals are now cleared from the equation. Next, isolate all
terms containing the variable x on one side of the equation. Remove −16x
from the right-hand side by adding 16x to both sides of the equation.
17x + 21 + 16x = −16x + 25 + 16x
33x + 21 = 25
Add 16x to both sides.
Simplify both sides.
Subtract 21 from both sides to remove the term 21 from the left-hand side of
the equation.
33x + 21 − 21 = 25 − 21
33x = 4
Subtract 21 from both sides.
Simplify both sides.
Finally, to “undo” multiplying by 33, divide both sides of the equation by 33.
33x
4
=
33
33
4
x=
33
Second Edition: 2012-2013
Divide both sides by 33.
Simplify.