Download Solving Equations by the Multiplication Property

2.2
Solving Equations by the
Multiplication Property
2.2
OBJECTIVES
1. Determine whether a given number is a solution
for an equation
2. Use the multiplication property to solve
equations
3. Find the mean for a given set
Let’s look at a different type of equation. For instance, what if we want to solve an equation
like the following?
6x 18
Using the addition property of the last section won’t help. We will need a second property
for solving equations.
Rules and Properties: The Multiplication Property of Equality
If a b
NOTE Again, as long as you do
the same thing to both sides of
the equation, the “balance” is
maintained.
NOTE Do you see why the
number cannot be 0?
Multiplying by 0 gives 0 0.
We have lost the variable!
then
ac bc
where c 0
In words, multiplying both sides of an equation by the same nonzero number
gives an equivalent equation.
Again, we return to the image of the balance scale. We start with the assumption that a and
b have the same weight.
a
b
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The multiplication property tells us that the scale will be in balance as long as we have the
same number of “a weights” as we have of “b weights.”
a a
aaa
b b
bbb
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156
CHAPTER 2
EQUATIONS AND INEQUALITIES
Let’s work through some examples, using this second rule.
Example 1
Solving Equations by Using the Multiplication Property
Solve
6x 18
Here the variable x is multiplied by 6. So we apply the multiplication property and multi1
ply both sides by . Keep in mind that we want an equation of the form
6
x
NOTE
1
1
(6x) 6 x
6
6
1 x, or x
1
1
(6x) 18
6
6
We then have x alone on the
left, which is what we want.
We can now simplify.
1x3
or
x3
The solution is 3. To check, replace x with 3:
6 3 18
18 18
(True)
CHECK YOURSELF 1
Solve and check.
8x 32
In Example 1 we solved the equation by multiplying both sides by the reciprocal of the
coefficient of the variable.
Example 2 illustrates a slightly different approach to solving an equation by using the
multiplication property.
Example 2
Solving Equations by Using the Multiplication Property
5x 35
NOTE Because division is
The variable x is multiplied by 5. We divide both sides by 5 to “undo” that multiplication:
defined in terms of
multiplication, we can also
divide both sides of an
equation by the same nonzero
number.
5x
35
5
5
x 7
Note that the right side
reduces to 7. Be careful
with the rules for signs.
We will leave it to you to check the solution.
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Solve
SOLVING EQUATIONS BY THE MULTIPLICATION PROPERTY
SECTION 2.2
157
CHECK YOURSELF 2
Solve and check.
7x 42
Example 3
Solving Equations by Using the Multiplication Property
Solve
9x 54
In this case, x is multiplied by 9, so we divide both sides by 9 to isolate x on the left:
9x
54
9
9
x 6
The solution is 6. To check:
(9)(6) 54
54 54
(True)
CHECK YOURSELF 3
Solve and check.
10x 60
Example 4 illustrates the use of the multiplication property when fractions appear in an
equation.
Example 4
Solving Equations by Using the Multiplication Property
(a) Solve
x
6
3
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Here x is divided by 3. We will use multiplication to isolate x.
3
3 3 6
x
x 18
To check:
18
6
3
66
(True)
This leaves x alone on the
left because
x
3 x
x
x
3
3
1 3
1
CHAPTER 2
EQUATIONS AND INEQUALITIES
(b) Solve
x
9
5
5
5 5(9)
x
Because x is divided by 5,
multiply both sides by 5
x 45
The solution is 45. To check, we replace x with 45:
45
9
5
9 9
(True)
The solution is verified.
CHECK YOURSELF 4
Solve and check.
x
3
7
(a)
(b)
x
8
4
When the variable is multiplied by a fraction that has a numerator other than 1, there are
two approaches to finding the solution.
Example 5
Solving Equations by Using Reciprocals
Solve
3
x9
5
One approach is to multiply by 5 as the first step.
5
5 x 5 9
3
3x 45
Now we divide by 3.
3x
45
3
3
x 15
To check:
3
15 9
5
99
(True)
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158
SOLVING EQUATIONS BY THE MULTIPLICATION PROPERTY
SECTION 2.2
159
A second approach combines the multiplication and division steps and is generally a bit
5
more efficient. We multiply by .
3
5
is the
3
3
reciprocal of , and the product
5
of a number and its reciprocal is
just 1! So
NOTE Recall that
35 1
5
3
5 3
5
x 9
3 5
3
3
5 9
x 15
3
1
1
So x 15, as before.
CHECK YOURSELF 5
Solve and check.
2
x 18
3
You may sometimes have to simplify an equation before applying the methods of this
section. Example 6 illustrates this property.
Example 6
Combining Like Terms and Solving Equations
Solve and check:
3x 5x 40
Using the distributive property, we can combine the like terms on the left to write
8x 40
We can now proceed as before.
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8x
40
8
8
Divide by 8.
x5
The solution is 5. To check, we return to the original equation. Substituting 5 for x yields
3 5 5 5 40
15 25 40
40 40
(True)
The solution is verified.
CHAPTER 2
EQUATIONS AND INEQUALITIES
CHECK YOURSELF 6
Solve and check.
7x 4x 66
An average is a value that is representative of a set of numbers. One kind of average is
the mean.
Definitions: Mean
The mean of a set is the sum of the set divided by the number of elements in the
set. The mean is written as x (sometimes called “x-bar”). In mathematical
symbols, we say
x
x
n
The sum of the set
The number of elements in the set
Example 7
Finding the Mean
Find the mean for each set of numbers.
(a) 2, 3, 5, 4, 7
We begin by finding x.
x 2 (3) 5 4 7 15
Next we find n.
n5
Remember that n is the number of elements in the set.
Finally, we substitute our numbers into the equation.
x
x
15
3
n
5
The mean of the set is 3.
(b) 4, 7, 9, 3, 6, 2, 3, 8
First find x.
x (4) 7 9 (3) 6 (2) (3) 8 18
Next find n.
n8
Substitute these numbers into the equation
x
x
18
9
(or 2.25)
n
8
4
The mean of this set is
9
or 2.25
4
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160
SOLVING EQUATIONS BY THE MULTIPLICATION PROPERTY
SECTION 2.2
161
CHECK YOURSELF 7
Find the mean for each set of numbers.
(a) 5, 2, 6, 3, 2
(b) 6, 2, 3, 8, 5, 6, 1, 3
Example 8
Finding the Mean
During a week in February the low temperature in Fargo, North Dakota, was recorded
each day. The results are presented in the following table. Find both the median and the
mean for the set of numbers.
M
T
W
Th
F
Sa
Su
11
17
15
18
20
2
20
110
100
90
80
70
60
50
40
30
20
10
0
–10
–20
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NOTE You can review the
discussion of the median in
Section 1.3.
To find the median we place the numbers in ascending order:
20
18
17
15
11
2
20
The median is the middle value, so the median is 15 degrees.
To find the mean, we first find x.
x (11) (17) (15) (18) (20) (2) 20 63
Then, given that n 7, we use the equation for the mean.
x
x
63
9
n
7
The mean is 9.
CHAPTER 2
EQUATIONS AND INEQUALITIES
Which average was more appropriate? There is really no “right” answer to that question.
In this case, the median would probably be preferred by most statisticians. It yields a temperature that was actually the low temperature on Wednesday of that week, so it is more
representative of the set of low temperatures.
CHECK YOURSELF 8
The low temperatures in Anchorage, Alaska, for one week in January are given in
the following table. Compute both the median and the mean low temperature for
that week.
M
T
6
10
W
Th
12
22
F
28
Sa
Su
26
27
CHECK YOURSELF ANSWERS
1. 4
2. 6
7. (a) 2; (b) 1.5
3. 6
4. (a) 21; (b) 32
5. 27
8. mean 17, median 22
6. 6
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162
Name
2.2
Exercises
Section
Date
Solve for x and check your result.
ANSWERS
1. 5x 20
4. 6x 42
7. 4x 16
10. 10x 100
13. 4x 12
16. 7x 35
19.
22.
8. 3x 27
11. 6x 54
14. 52 4x
17. 6x 54
20.
x
5
8
23. 6 x
2
3
x
7
x
26.
5
7
3. 9x 54
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
x
3
5
25.
26.
27.
28.
x
3
29.
30.
31.
32.
33.
34.
35.
36.
6. 66 6x
9. 9x 72
12. 7x 49
15. 42 6x
18. 4x 24
21.
24. 6 x
27. 8
3
x
3
4
29.
2
x 0.9
3
30.
3
x 15
4
32.
3
6
x 10 5
5
33. x 15
28. © 2001 McGraw-Hill Companies
5. 63 9x
x
4
2
x
25.
4
5
31.
2. 6x 30
34. 5x 4x 36
35. 16x 9x 16.1
4
x8
5
5
6
36. 4x 2x 7x 36
163
ANSWERS
37.
Once again, certain equations involving decimal fractions can be solved by the methods
of this section. For instance, to solve 2.3x 6.9 we simply use our multiplication
property to divide both sides of the equation by 2.3. This will isolate x on the left as
desired. Use this idea to solve each of the following equations for x.
38.
39.
40.
37. 3.2x 12.8
38. 5.1x 15.3
39. 4.5x 3.51
40. 8.2x 31.078
41. 1.3x 2.8x 12.3
42. 2.7x 5.4x 16.2
41.
42.
43.
44.
Find the median and the mean of each data set.
45.
46.
47.
48.
43. 2, 3, 4, 5, 6
44. 1, 3, 8, 10, 18
45. 3, 1, 2, 4, 6, 10
46. 5, 2, 1, 4, 6, 8
3
2
47. , 1, 2,
5
, 3, 7
2
4
3
1 2
, 5, 6
3 3
48. , ,
49.
49. Average weight. Kareem bought four bags of candy. The weights of the bags were
50.
16 ounces (oz), 21 oz, 18 oz, and 15 oz. Find the median and the mean weight of the
bags of candy.
a.
b.
50. Average savings. Jose has savings accounts for each of his five children. They
contain $215, $156, $318, $75, and $25. Find the median and the mean amount of
money per account.
c.
d.
e.
Getting Ready for Section 2.3 [Section 1.2]
f.
(a) 2(x 3)
(e) 7(3x 4)
h.
(b) 3(a 4)
(f) 4(5x 4)
(c) 5(2b 1)
(g) 3(4x 3)
(d) 3(3p 4)
(h) 5(3y 2)
Answers
1. 4
3. 6
5. 7
7. 4
9. 8
11. 9
13. 3
15. 7
17. 9
19. 8
21. 15
23. 42
25. 20
27. 24
29. 1.35
31. 20
33. 18
35. 2.3
37. 4
39. 0.78
41. 3
43. 4
45. 3
47. 2
49. Mean: 17.5, Median: 17 oz.
a. 2x 6
b. 3a 12
c. 10b 5
d. 9p 12
e. 21x 28
f. 20x 16
g. 12x 9
h. 15y 10
164
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Use the distributive property to remove the parentheses in the following expressions.
g.