Download Algebraic Expressions and Equations: Solving

OpenStax-CNX module: m35045
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Algebraic Expressions and
Equations: Solving Equations of the
∗
Form ax=b and x/a=b
Wade Ellis
Denny Burzynski
This work is produced by OpenStax-CNX and licensed under the
Creative Commons Attribution License 3.0
†
Abstract
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr.
module discusses solving equations of the form
ax = b
and
x
a
= b.
This
By the end of the module students
should be familiar with the multiplication/division property of equality, be able to solve equations of the
form ax
=b
and
x
a
=b
and be able to use combined techniques to solve equations.
1 Section Overview
• Multiplication/ Division Property of Equality
• Combining Techniques in Equations Solving
2 Multiplication/ Division Property of Equality
Recall that the equal sign of an equation indicates that the number represented by the expression on the left
side is the same as the number represented by the expression on the right side. From this, we can suggest
the multiplication/division property of equality.
Multiplication/Division Property of Equality
Given any equation,
1. We can obtain an equivalent equation by multiplying both sides of the equation by the same nonzero
number, that is, if c 6= 0, then a = b is equivalent to
a·c=b·c
2. We can obtain an equivalent equation by dividing both sides of the equation by the same nonzero
number, that is, if c 6= 0, then a = b is equivalent to
a
c
=
b
c
The multiplication/division property of equality can be used to undo an association with a number that
multiplies or divides the variable.
∗ Version
1.3: Aug 23, 2010 10:34 am -0500
† http://creativecommons.org/licenses/by/3.0/
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2.1 Sample Set A
Use the multiplication / division property of equality to solve each equation.
Example 1
6y = 54
6 is associated with y by multiplication. Undo the association by dividing both sides by 6
6y
6
=
54
6
9
)6y
)6
=
)54
)6
y=9
Check: When y = 9
6y = 54
becomes
,
a true statement.
The solution to 6y = 54 is y = 9.
Example 2
= 27.
-2 is associated with x by division. Undo the association by multiplying both sides by -2.
x
(−2) −2
= (−2) 27
x
−2
) − 2 x = (−2) 27
)−2
x = −54
Check: When x = −54,
x
−2 = 27
becomes
a true statement.
The solution to
x
−2
= 27 is x = −54
Example 3
= 6.
We will examine two methods for solving equations such as this one.
Method 1: Use of dividing out common factors.
3a
7
3a
7
=6
7·
3a
7
7 is associated with a by division. Undo the association by multiplying both sides by 7.
=7·6
Divide out the 7's.
)7 · 3a = 42
)7
3a = 42
3 is associated with a by multiplication. Undo the association by dviding both sides by 3.
3a
42
3 = 3
)3a
= 14
)3
a = 14
Check: When a = 14,
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OpenStax-CNX module: m35045
3a
7
3
=6
becomes
,
a true statement.
The solution to 3a
7 = 6 is a = 14.
Method 2: Use of reciprocals
Recall that if the product of two numbers is 1, the numbers are reciprocals. Thus
reciprocals.
3a
7
=6
7
3
3a
7
3
7
and
7
3
are
Multiply both sides of the equation by 73 , the reciprocal of 37 .
·
1
)7
)3
)3a
)7
·
1
7
3
=
1
·6
2
=
7
)3
1
·
)6
1
1
1 · a = 14
a = 14
Notice that we get the same solution using either method.
Example 4
−8x = 24
-8 is associated with x by multiplication. Undo the association by dividing both sides by -8.
−8x
−8
=
24
−8
−8x
−8
24
= −8
x = −3
Check: When x = −3,
−8x = 24
becomes
a true statement.
,
Example 5
−x = 7.
Since − − x is actually −1 · x and (−1) (−1) = 1, we can isolate x by multiplying both sides of the
equation by −1.
(−1) (−x) = −1 · 7
x = −7
Check: When x = 7,
−x = 7
becomes
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The solution to −x = 7 is x = −7.
2.2 Practice Set A
Use the multiplication/division property of equality to solve each equation. Be sure to check each solution.
Exercise 1
(Solution on p. 9.)
Exercise 2
(Solution on p. 9.)
Exercise 3
(Solution on p. 9.)
7x = 21
−5x = 65
x
4
= −8
Exercise 4
3x
8
(Solution on p. 9.)
=6
Exercise 5
(Solution on p. 9.)
−y = 3
Exercise 6
(Solution on p. 9.)
−k = −2
3 Combining Techniques in Equation Solving
Having examined solving equations using the addition/subtraction and the multiplication/division principles
of equality, we can combine these techniques to solve more complicated equations.
When beginning to solve an equation such as 6x − 4 = −16, it is helpful to know which property of
equality to use rst, addition/subtraction or multiplication/division. Recalling that in equation solving we
are trying to isolate the variable (disassociate numbers from it), it is helpful to note the following.
To associate numbers and letters, we use the order of operations.
1. Multiply/divide
2. Add/subtract
To undo an association between numbers and letters, we use the order of operations in reverse.
1. Add/subtract
2. Multiply/divide
3.1 Sample Set B
Solve each equation. (In these example problems, we will not show the checks.)
Example 6
6x − 4 = −16
-4 is associated with x by subtraction. Undo the association by adding 4 to both sides.
6x − 4 + 4 = −16 + 4
6x = −12
6 is associated with x by multiplication. Undo the association by dividing both sides by 6
= −612
x = −2
6x
6
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OpenStax-CNX module: m35045
Example 7
−8k + 3= − 45.
3 is associated with k by addition. Undo the association by subtracting 3 from both sides.
−8k + 3 − 3= − 45 − 3
−8k = − 48
-8 is associated with k by multiplication. Undo the association by dividing both sides by -8.
−8k
−8
=
k=6
−48
−8
Example 8
5m − 6 − 4m = 4m − 8 + 3m. Begin by solving this equation by combining like terms.
m − 6 = 7m − 8 Choose a side on which to isolate m. Since 7 is greater than 1, we'll isolate m
on the right side.
Subtract m from both sides.
−m − 6 − m = 7m − 8 − m
−6 = 6m − 8
8 is associated with m by subtraction. Undo the association by adding 8 to both sides.
−6 + 8 = 6m − 8 + 8
2 = 6m
6 is associated with m by multiplication. Undo the association by dividing both sides by 6.
6m
2
6 = 6 Reduce.
1
3 =m
Notice that if we had chosen to isolate m on the left side of the equation rather than the right
side, we would have proceeded as follows:
m − 6 = 7m − 8
Subtract 7m from both sides.
m − 6 − 7m = 7m − 8 − 7m
−6m − 6= − 8
Add 6 to both sides,
−6m − 6 + 6= − 8 + 6
−6m= − 2
Divide both sides by -6.
−6m
−2
−6 = −6
m = 13
This is the same result as with the previous approach.
Example 9
8x
7 =
−2
7 is associated with x by division. Undo the association by multiplying both sides by 7.
8x
= 7 (−2)
)7
8x
7 = − 14
)7 ·
7·
8x= − 14
8 is associated with x by multiplication. Undo the association by dividing both sides by 8.
)8x
)8
x=
−7
4
−7
4
=
3.2 Practice Set B
Solve each equation. Be sure to check each solution.
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Exercise 7
(Solution on p. 9.)
Exercise 8
(Solution on p. 9.)
5m + 7= − 13
−3a − 6 = 9
Exercise 9
(Solution on p. 9.)
Exercise 10
(Solution on p. 9.)
Exercise 11
(Solution on p. 9.)
2a + 10 − 3a = 9
11x − 4 − 13x = 4x + 14
−3m + 8 = −5m + 1
Exercise 12
5y + 8y − 11= − 11
(Solution on p. 9.)
4 Exercises
Solve each equation. Be sure to check each result.
Exercise 13
7x = 42
(Solution on p. 9.)
Exercise 14
8x = 81
Exercise 15
10x = 120
(Solution on p. 9.)
Exercise 16
11x = 121
Exercise 17
−6a = 48
(Solution on p. 9.)
Exercise 18
−9y = 54
Exercise 19
−3y = − 42
(Solution on p. 9.)
Exercise 20
−5a= − 105
Exercise 21
2m= − 62
(Solution on p. 9.)
Exercise 22
3m= − 54
Exercise 23
x
4
(Solution on p. 9.)
=7
Exercise 24
y
3
= 11
Exercise 25
−z
6 =
− 14
Exercise 26
−w
5
=1
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(Solution on p. 9.)
OpenStax-CNX module: m35045
Exercise 27
3m − 1= − 13
7
(Solution on p. 9.)
Exercise 28
4x + 7= − 17
Exercise 29
2 + 9x= − 7
(Solution on p. 9.)
Exercise 30
5 − 11x = 27
Exercise 31
32 = 4y + 6
(Solution on p. 9.)
Exercise 32
−5 + 4 = −8m + 1
Exercise 33
3k + 6 = 5k + 10
(Solution on p. 9.)
Exercise 34
4a + 16 = 6a + 8a + 6
Exercise 35
6x + 5 + 2x − 1 = 9x − 3x + 15
(Solution on p. 9.)
Exercise 36
−9y − 8 + 3y + 7 = −7y + 8y − 5y + 9
Exercise 37
(Solution on p. 9.)
−3a = a + 5
Exercise 38
5b= − 2b + 8b + 1
Exercise 39
−3m + 2 − 8m − 4 = −14m + m − 4
(Solution on p. 10.)
Exercise 40
5a + 3 = 3
Exercise 41
(Solution on p. 10.)
7x + 3x = 0
Exercise 42
7g + 4 − 11g = −4g + 1 + g
Exercise 43
5a
7
= 10
(Solution on p. 10.)
Exercise 44
2m
9
=4
Exercise 45
3x
4
=
(Solution on p. 10.)
9
2
Exercise 46
8k
3
= 32
Exercise 47
3a
8
−
3
2
=0
Exercise 48
5m
6
−
25
3
=0
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(Solution on p. 10.)
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4.1 Exercises for Review
Exercise 49
() Use the distributive property to compute 40 · 28.
(Solution on p. 10.)
Exercise 50
() Approximating π by 3.14, nd the approximate circumference of the circle.
Exercise 51
() Find the area of the parallelogram.
Exercise 52
() Find the value of
(Solution on p. 10.)
−3(4−15)−2
.
−5
Exercise 53
() Solve the equation x − 14 + 8 = −2.
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(Solution on p. 10.)
OpenStax-CNX module: m35045
Solutions to Exercises in this Module
Solution to Exercise (p. 4)
x=3
Solution to Exercise (p. 4)
x = −13
Solution to Exercise (p. 4)
x = −32
Solution to Exercise (p. 4)
x = 16
Solution to Exercise (p. 4)
y = −3
Solution to Exercise (p. 4)
k=2
Solution to Exercise (p. 5)
m= − 4
Solution to Exercise (p. 6)
a= − 5
Solution to Exercise (p. 6)
a=1
Solution to Exercise (p. 6)
x= − 3
Solution to Exercise (p. 6)
m= −
7
2
Solution to Exercise (p. 6)
y=0
Solution to Exercise (p. 6)
x=6
Solution to Exercise (p. 6)
x = 12
Solution to Exercise (p. 6)
a= − 8
Solution to Exercise (p. 6)
y = 14
Solution to Exercise (p. 6)
m= − 31
Solution to Exercise (p. 6)
x = 28
Solution to Exercise (p. 6)
z = 84
Solution to Exercise (p. 7)
m = −4
Solution to Exercise (p. 7)
x = −1
Solution to Exercise (p. 7)
y=
13
2
Solution to Exercise (p. 7)
k= − 2
Solution to Exercise (p. 7)
x=
11
2
or 5 21
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Solution to Exercise (p. 7)
a = − 54
Solution to Exercise (p. 7)
m= − 1
Solution to Exercise (p. 7)
x=0
Solution to Exercise (p. 7)
a = 14
Solution to Exercise (p. 7)
x=6
Solution to Exercise (p. 7)
a=4
Solution to Exercise (p. 8)
40 (30 − 2) = 1200 − 80 = 1120
Solution to Exercise (p. 8)
220 sq cm
Solution to Exercise (p. 8)
x=4
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