Solving Equations One Step

Chapter
2
Solving Linear Equations
2.1
Solving Equations: One Step
1. Substitute 2 for x in the equation, then simplify both sides of the resulting
equation.
x+2=4
Original equation.
2+2=4
Substitute 2 for x.
4=4
Simplify both sides.
This last equation is a true statement. Hence, 2 is a solution of the equation
x + 2 = 4. For contrast, substitute 3 for x in the equation and simplify.
x+2=4
Original equation.
3+2=4
5=4
Substitute 3 for x.
Simplify both sides.
This last equation is not a true statement. Hence, 3 is not a solution of
x + 2 = 4. Readers should check that the remaining two given numbers are
not solutions by substituting them into the equation and showing that a false
statement results.
3. Substitute 13 for x in the equation, then simplify both sides of the resulting
equation.
x−9=4
13 − 9 = 4
Original equation.
Substitute 13 for x.
4=4
Simplify both sides.
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CHAPTER 2. SOLVING LINEAR EQUATIONS
54
This last equation is a true statement. Hence, 13 is a solution of the equation
x − 9 = 4. For contrast, substitute 14 for x in the equation and simplify.
x−9=4
14 − 9 = 4
Original equation.
Substitute 14 for x.
5=4
Simplify both sides.
This last equation is not a true statement. Hence, 14 is not a solution of
x − 9 = 4. Readers should check that the remaining two given numbers are
not solutions by substituting them into the equation and showing that a false
statement results.
5. Substitute 9 for x in the equation, then simplify both sides of the resulting
equation.
x−3= 6
9−3= 6
6=6
Original equation.
Substitute 9 for x.
Simplify both sides.
This last equation is a true statement. Hence, 9 is a solution of the equation
x − 3 = 6. For contrast, substitute 10 for x in the equation and simplify.
x−3=6
10 − 3 = 6
Original equation.
Substitute 10 for x.
7=6
Simplify both sides.
This last equation is not a true statement. Hence, 10 is not a solution of
x − 3 = 6. Readers should check that the remaining two given numbers are
not solutions by substituting them into the equation and showing that a false
statement results.
7. The number −6 is the only solution of the equation x − 1 = −7. Similarly,
−8 is the only solution of the equation x = −8. Therefore x − 1 = −7 and
x = −8 do not have the same solution sets and are not equivalent.
9. The number 0 is the only solution of the equation x − 5 = −5. Similarly,
0 is the only solution of the equation x = 0. Therefore x − 5 = −5 and x = 0
have the same solution sets and are equivalent.
11. By inspection, the equation x2 = 1 has two solutions, −1 and 1. On the
other hand, the equation x = 1 has a single solution, namely 1. Hence the
equations x2 = 1 and x = 1 do not have the same solution sets and are not
equivalent.
Second Edition: 2012-2013
2.1. SOLVING EQUATIONS: ONE STEP
55
13. To undo the effect of subtracting 20, add 20 to both sides of the equation.
x − 20 = 9
x − 20 + 20 = 9 + 20
x = 29
Original equation.
Add 20 to both sides.
On the left, adding 20 “undoes” the
effect of subtracting 20 and returns x.
On the right, 9 + 20 = 29.
Hence, 29 is a solution of x − 20 = 9.
15. To undo the effect of subtracting 3, add 3 to both sides of the equation.
16 = x − 3
16 + 3 = x − 3 + 3
19 = x
Original equation.
Add 3 to both sides.
On the right, adding 3 “undoes” the
effect of subtracting 3 and returns x.
On the left, 16 + 3 = 19.
Hence, 19 is a solution of 16 = x − 3.
17. To undo the effect of adding 11, subtract 11 from both sides of the equation.
x + 11 = 20
x + 11 − 11 = 20 − 11
x=9
Original equation.
Subtract 11 from both sides.
On the left, subtracting 11 “undoes”
the effect of adding 11 and returns x.
On the right, 20 − 11 = 9.
Hence, 9 is a solution of x + 11 = 20.
19. To undo the effect of subtracting 19, add 19 to both sides of the equation.
9 = x − 19
9 + 19 = x − 19 + 19
28 = x
Original equation.
Add 19 to both sides.
On the right, adding 19 “undoes” the
effect of subtracting 19 and returns x.
On the left, 9 + 19 = 28.
Hence, 28 is a solution of 9 = x − 19.
Second Edition: 2012-2013
CHAPTER 2. SOLVING LINEAR EQUATIONS
56
21. To undo the effect of adding 9, subtract 9 from both sides of the equation.
20 = 9 + x
20 − 9 = 9 + x − 9
11 = x
Original equation.
Subtract 9 from both sides.
On the right, subtracting 9 “undoes”
the effect of adding 9 and returns x.
On the left, 20 − 9 = 11.
Hence, 11 is a solution of 20 = 9 + x.
23. To undo the effect of adding 17, subtract 17 from both sides of the equation.
18 = 17 + x
18 − 17 = 17 + x − 17
1=x
Original equation.
Subtract 17 from both sides.
On the right, subtracting 17 “undoes”
the effect of adding 17 and returns x.
On the left, 18 − 17 = 1.
Hence, 1 is a solution of 18 = 17 + x.
25. To undo the effect of adding 7, subtract 7 from both sides of the equation.
7 + x = 19
7 + x − 7 = 19 − 7
x = 12
Original equation.
Subtract 7 from both sides.
On the left, subtracting 7 “undoes”
the effect of adding 7 and returns x.
On the right, 19 − 7 = 12.
Hence, 12 is a solution of 7 + x = 19.
27. To undo the effect of subtracting 9, add 9 to both sides of the equation.
x−9=7
x−9+9=7+9
x = 16
Original equation.
Add 9 to both sides.
On the left, adding 9 “undoes” the
effect of subtracting 9 and returns x.
On the right, 7 + 9 = 16.
Hence, 16 is a solution of x − 9 = 7.
Second Edition: 2012-2013
2.1. SOLVING EQUATIONS: ONE STEP
57
29. To undo the effect of adding 15, subtract 15 from both sides of the equation.
x + 15 = 19
x + 15 − 15 = 19 − 15
x=4
Original equation.
Subtract 15 from both sides.
On the left, subtracting 15 “undoes”
the effect of adding 15 and returns x.
On the right, 19 − 15 = 4.
Hence, 4 is a solution of x + 15 = 19.
31. To undo the effect of adding 10, subtract 10 from both sides of the equation.
10 + x = 15
10 + x − 10 = 15 − 10
x=5
Original equation.
Subtract 10 from both sides.
On the left, subtracting 10 “undoes”
the effect of adding 10 and returns x.
On the right, 15 − 10 = 5.
Hence, 5 is a solution of 10 + x = 15.
33. To undo the effects of subtracting 4/9, first add 4/9 to both sides of the
equation. Then make equivalent fractions with a common denominator and
simplify.
4
2
=
9
7
4 4
2 4
x− + = +
9 9
7 9
18 28
+
x=
63 63
x−
x=
46
63
Original equation.
Add 4/9 to both sides.
On the left, adding 4/9 “undoes”
the effect of subtracting 4/9 and returns x.
On the right, make equivalent fractions
with a common denominator.
Simplify.
35. To undo the effects of adding 7/4, first subtract 7/4 from both sides of the
equation. Then make equivalent fractions with a common denominator and
Second Edition: 2012-2013
CHAPTER 2. SOLVING LINEAR EQUATIONS
58
simplify.
7
4
=−
4
9
7 7
4 7
x+ − =− −
4 4
9 4
16 63
x=− −
36 36
x+
−79
36
x=
Original equation.
Subtract 7/4 from both sides.
On the left, subtracting 7/4 “undoes”
the effect of adding 7/4 and returns x.
On the right, make equivalent fractions
with a common denominator.
Simplify.
37. To undo the effects of adding 5/9, first subtract 5/9 from both sides of the
equation. Then make equivalent fractions with a common denominator and
simplify.
5
7
=
9
2
5 5
7 5
x+ − = −
9 9
2 9
63 10
x=
−
18 18
x+
x=
53
18
Original equation.
Subtract 5/9 from both sides.
On the left, subtracting 5/9 “undoes”
the effect of adding 5/9 and returns x.
On the right, make equivalent fractions
with a common denominator.
Simplify.
39. To undo the effects of subtracting 9/8, first add 9/8 to both sides of the
equation. Then make equivalent fractions with a common denominator and
simplify.
9
1
=−
8
2
9 9
1 9
x− + =− +
8 8
2 8
4 9
x=− +
8 8
x−
x=
5
8
Second Edition: 2012-2013
Original equation.
Add 9/8 to both sides.
On the left, adding 9/8 “undoes”
the effect of subtracting 9/8 and returns x.
On the right, make equivalent fractions
with a common denominator.
Simplify.
2.1. SOLVING EQUATIONS: ONE STEP
59
41. To undo multiplying by −5.1, divide both sides of the equation by −5.1.
−5.1x = −12.75
−5.1x
−12.75
=
−5.1
−5.1
x = 2.5
Original equation.
Divide both sides by −5.1.
Simplify: −12.75/(−5.1) = 2.5.
43. To undo multiplying by −6.9, divide both sides of the equation by −6.9.
−6.9x = −58.65
−6.9x
−58.65
=
−6.9
−6.9
x = 8.5
Original equation.
Divide both sides by −6.9.
Simplify: −58.65/(−6.9) = 8.5.
45. To undo multiplying by −3.6, divide both sides of the equation by −3.6.
−3.6x = −24.12
−3.6x
−24.12
=
−3.6
−3.6
x = 6.7
Original equation.
Divide both sides by −3.6.
Simplify: −24.12/(−3.6) = 6.7.
47. To undo the effect of dividing by 2, multiply both sides of the equation by
2.
x
= −11
x 2
2
= 2 (−11)
2
x = −22
Original equation.
Multiply both sides by 2.
On the left, simplify. On the
right, multiply: 2(−11) = −22.
49. To undo the effect of dividing by 8, multiply both sides of the equation by
8.
x
= −18
x 8
8
= 8 (−18)
8
x = −144
Original equation.
Multiply both sides by 8.
On the left, simplify. On the
right, multiply: 8(−18) = −144.
Second Edition: 2012-2013