<< Chapter < Page Chapter >> Page >

Solve: −11 = 1 2 ( 6 p + 2 ) .

p = −4

Got questions? Get instant answers now!

Solve: 8 = 1 3 ( 9 q + 6 ) .

q = 2

Got questions? Get instant answers now!

In the next example, even after distributing, we still have fractions to clear.

Solve: 1 2 ( y 5 ) = 1 4 ( y 1 ) .

Solution

.
Distribute. .
Simplify. .
Multiply by the LCD, 4. .
Distribute. .
Simplify. .
Collect the variables to the left. .
Simplify. .
Collect the constants to the right. .
Simplify. .
Check: .
Let y = 9 . .
Finish the check on your own.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve: 1 5 ( n + 3 ) = 1 4 ( n + 2 ) .

n = 2

Got questions? Get instant answers now!

Solve: 1 2 ( m 3 ) = 1 4 ( m 7 ) .

m = −1

Got questions? Get instant answers now!

Solve: 5 x 3 4 = x 2 .

Solution

.
Multiply by the LCD, 4. .
Simplify. .
Collect the variables to the right. .
Simplify. .
Divide. .
Simplify. .
Check: .
Let x = 1 . .
.
.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve: 4 y 7 3 = y 6 .

y = 2

Got questions? Get instant answers now!

Solve: −2 z 5 4 = z 8 .

z = −2

Got questions? Get instant answers now!

Solve: a 6 + 2 = a 4 + 3 .

Solution

.
Multiply by the LCD, 12. .
Distribute. .
Simplify. .
Collect the variables to the right. .
Simplify. .
Collect the constants to the left. .
Simplify. .
Check: .
Let a = −12 . .
.
.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve: b 10 + 2 = b 4 + 5 .

b = −20

Got questions? Get instant answers now!

Solve: c 6 + 3 = c 3 + 4 .

c = −6

Got questions? Get instant answers now!

Solve: 4 q + 3 2 + 6 = 3 q + 5 4 .

Solution

.
Multiply by the LCD, 4. .
Distribute. .
Simplify. .
.
.
Collect the variables to the left. .
Simplify. .
Collect the constants to the right. .
Simplify. .
Divide by 5. .
Simplify. .
Check: .
Let q = −5 . .
Finish the check on your own.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve: 3 r + 5 6 + 1 = 4 r + 3 3 .

r = 1

Got questions? Get instant answers now!

Solve: 2 s + 3 2 + 1 = 3 s + 2 4 .

s = −8

Got questions? Get instant answers now!

Solve equations with decimal coefficients

Some equations have decimals in them. This kind of equation will occur when we solve problems dealing with money or percentages. But decimals can also be expressed as fractions. For example, 0.3 = 3 10 and 0.17 = 17 100 . So, with an equation with decimals, we can use the same method we used to clear fractions—multiply both sides of the equation by the least common denominator.

Solve: 0.06 x + 0.02 = 0.25 x 1.5 .

Solution

Look at the decimals and think of the equivalent fractions.

0.06 = 6 100 0.02 = 2 100 0.25 = 25 100 1.5 = 1 5 10

Notice, the LCD is 100.

By multiplying by the LCD, we will clear the decimals from the equation.

.
Multiply both sides by 100. .
Distribute. .
Multiply, and now we have no more decimals. .
Collect the variables to the right. .
Simplify. .
Collect the constants to the left. .
Simplify. .
Divide by 19. .
Simplify. .
Check: Let x = 8 .
.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve: 0.14 h + 0.12 = 0.35 h 2.4 .

h = 12

Got questions? Get instant answers now!

Solve: 0.65 k 0.1 = 0.4 k 0.35 .

k = −1

Got questions? Get instant answers now!

The next example uses an equation that is typical of the money applications in the next chapter. Notice that we distribute the decimal before we clear all the decimals.

Solve: 0.25 x + 0.05 ( x + 3 ) = 2.85 .

Solution

.
Distribute first. .
Combine like terms. .
To clear decimals, multiply by 100. .
Distribute. .
Subtract 15 from both sides. .
Simplify. .
Divide by 30. .
Simplify. .
Check it yourself by substituting x = 9 into the original equation.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve: 0.25 n + 0.05 ( n + 5 ) = 2.95 .

n = 9

Got questions? Get instant answers now!

Solve: 0.10 d + 0.05 ( d 5 ) = 2.15 .

d = 16

Got questions? Get instant answers now!

Key concepts

  • Strategy to Solve an Equation with Fraction Coefficients
    1. Find the least common denominator of all the fractions in the equation.
    2. Multiply both sides of the equation by that LCD. This clears the fractions.
    3. Solve using the General Strategy for Solving Linear Equations.

Practice makes perfect

Solve Equations with Fraction Coefficients

In the following exercises, solve each equation with fraction coefficients.

3 4 x 1 2 = 1 4

x = 1

Got questions? Get instant answers now!

5 6 y 1 3 = 7 6

y = −1

Got questions? Get instant answers now!

5 8 b + 1 2 = 3 4

b = −2

Got questions? Get instant answers now!

2 = 1 3 x 1 2 x + 2 3 x

Got questions? Get instant answers now!

2 = 3 5 x 1 3 x + 2 5 x

x = 3

Got questions? Get instant answers now!

1 4 m 4 5 m + 1 2 m = −1

Got questions? Get instant answers now!

5 6 n 1 4 n 1 2 n = −2

n = −24

Got questions? Get instant answers now!

x + 3 4 = 1 2 x 5 4

x = −4

Got questions? Get instant answers now!

3 2 z + 1 3 = z 2 3

z = −2

Got questions? Get instant answers now!

1 2 x 1 4 = 1 12 x + 1 6

Got questions? Get instant answers now!

1 2 a 1 4 = 1 6 a + 1 12

a = 1

Got questions? Get instant answers now!

1 3 b + 1 5 = 2 5 b 3 5

Got questions? Get instant answers now!

1 3 x + 2 5 = 1 5 x 2 5

x = −6

Got questions? Get instant answers now!

1 = 1 5 ( 15 x 10 )

x = 1

Got questions? Get instant answers now!

1 4 ( p 7 ) = 1 3 ( p + 5 )

Got questions? Get instant answers now!

1 5 ( q + 3 ) = 1 2 ( q 3 )

q = 7

Got questions? Get instant answers now!

1 3 ( x + 5 ) = 5 6

x = 5 2

Got questions? Get instant answers now!

4 m + 2 6 = m 3

m = −1

Got questions? Get instant answers now!

3 p + 6 3 = p 2

p = −4

Got questions? Get instant answers now!

v 10 + 1 = v 4 2

v = 20

Got questions? Get instant answers now!

d 6 + 3 = d 8 + 2

d = −24

Got questions? Get instant answers now!

3 x + 4 2 + 1 = 5 x + 10 8

Got questions? Get instant answers now!

10 y 2 3 + 3 = 10 y + 1 9

y = −1

Got questions? Get instant answers now!

7 u 1 4 1 = 4 u + 8 5

Got questions? Get instant answers now!

3 v 6 2 + 5 = 11 v 4 5

v = 4

Got questions? Get instant answers now!

Solve Equations with Decimal Coefficients

In the following exercises, solve each equation with decimal coefficients.

0.4 x + 0.6 = 0.5 x 1.2

Got questions? Get instant answers now!

0.7 x + 0.4 = 0.6 x + 2.4

x = 20

Got questions? Get instant answers now!

0.23 x + 1.47 = 0.37 x 1.05

Got questions? Get instant answers now!

0.48 x + 1.56 = 0.58 x 0.64

x = 22

Got questions? Get instant answers now!

0.9 x 1.25 = 0.75 x + 1.75

Got questions? Get instant answers now!

1.2 x 0.91 = 0.8 x + 2.29

x = 8

Got questions? Get instant answers now!

0.05 n + 0.10 ( n + 8 ) = 2.15

Got questions? Get instant answers now!

0.05 n + 0.10 ( n + 7 ) = 3.55

n = 19

Got questions? Get instant answers now!

0.10 d + 0.25 ( d + 5 ) = 4.05

Got questions? Get instant answers now!

0.10 d + 0.25 ( d + 7 ) = 5.25

d = 10

Got questions? Get instant answers now!

0.05 ( q 5 ) + 0.25 q = 3.05

Got questions? Get instant answers now!

0.05 ( q 8 ) + 0.25 q = 4.10

q = 15

Got questions? Get instant answers now!

Everyday math

Coins Taylor has $2.00 in dimes and pennies. The number of pennies is 2 more than the number of dimes. Solve the equation 0.10 d + 0.01 ( d + 2 ) = 2 for d , the number of dimes.

Got questions? Get instant answers now!

Stamps Paula bought $22.82 worth of 49-cent stamps and 21-cent stamps. The number of 21-cent stamps was 8 less than the number of 49-cent stamps. Solve the equation 0.49 s + 0.21 ( s 8 ) = 22.82 for s , to find the number of 49-cent stamps Paula bought.

s = 35

Got questions? Get instant answers now!

Writing exercises

Explain how you find the least common denominator of 3 8 , 1 6 , and 2 3 .

Got questions? Get instant answers now!

If an equation has several fractions, how does multiplying both sides by the LCD make it easier to solve?

Answers will vary.

Got questions? Get instant answers now!

If an equation has fractions only on one side, why do you have to multiply both sides of the equation by the LCD?

Got questions? Get instant answers now!

In the equation 0.35 x + 2.1 = 3.85 what is the LCD? How do you know?

100. Justifications will vary.

Got questions? Get instant answers now!

Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This is a table that has three rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can…,” “confidently,” “with some help,” and “no-I don’t get it!” The first column below “I can…” reads: “solve equations with fraction coefficients,” and “solve equations with decimal coefficients.” The rest of the cells are blank.

Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

Questions & Answers

Washing his dad’s car alone, eight-year-old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
Sam Reply
Ethan and Leo start riding their bikes at the opposite ends of a 65-mile bike path. After Ethan has ridden 1.5 hours and Leo has ridden 2 hours, they meet on the path. Ethan’s speed is 6 miles per hour faster than Leo’s speed. Find the speed of the two bikers.
Mckenzie Reply
Nathan walked on an asphalt pathway for 12 miles. He walked the 12 miles back to his car on a gravel road through the forest. On the asphalt he walked 2 miles per hour faster than on the gravel. The walk on the gravel took one hour longer than the walk on the asphalt. How fast did he walk on the gravel?
Mckenzie
Nancy took a 3 hour drive. She went 50 miles before she got caught in a storm. Then she drove 68 miles at 9 mph less than she had driven when the weather was good. What was her speed driving in the storm?
Reiley Reply
Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
hamzzi Reply
90 minutes
muhammad
Melody wants to sell bags of mixed candy at her lemonade stand. She will mix chocolate pieces that cost $4.89 per bag with peanut butter pieces that cost $3.79 per bag to get a total of twenty-five bags of mixed candy. Melody wants the bags of mixed candy to cost her $4.23 a bag to make. How many bags of chocolate pieces and how many bags of peanut butter pieces should she use?
Jake Reply
enrique borrowed $23,500 to buy a car he pays his uncle 2% interest on the $4,500 he borrowed from him and he pays the bank 11.5% interest on the rest. what average interest rate does he pay on the total $23,500
Nakiya Reply
13.5
Pervaiz
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost $8 per square foot and decorator tiles that cost $20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be $10 per square foot?
Bridget Reply
The equation P=28+2.54w models the relation between the amount of Randy’s monthly water bill payment, P, in dollars, and the number of units of water, w, used. Find the payment for a month when Randy used 15 units of water.
Bridget
help me understand graphs
Marlene Reply
what kind of graphs?
bruce
function f(x) to find each value
Marlene
I am in algebra 1. Can anyone give me any ideas to help me learn this stuff. Teacher and tutor not helping much.
Marlene
Given f(x)=2x+2, find f(2) so you replace the x with the 2, f(2)=2(2)+2, which is f(2)=6
Melissa
if they say find f(5) then the answer would be f(5)=12
Melissa
I need you to help me Melissa. Wish I can show you my homework
Marlene
How is f(1) =0 I am really confused
Marlene
what's the formula given? f(x)=?
Melissa
It shows a graph that I wish I could send photo of to you on here
Marlene
Which problem specifically?
Melissa
which problem?
Melissa
I don't know any to be honest. But whatever you can help me with for I can practice will help
Marlene
I got it. sorry, was out and about. I'll look at it now.
Melissa
Thank you. I appreciate it because my teacher assumes I know this. My teacher before him never went over this and several other things.
Marlene
I just responded.
Melissa
Thank you
Marlene
-65r to the 4th power-50r cubed-15r squared+8r+23 ÷ 5r
WENDY Reply
State the question clearly please
Rich
write in this form a/b answer should be in the simplest form 5%
August Reply
convert to decimal 9/11
August
0.81818
Rich
5/100 = .05 but Rich is right that 9/11 = .81818
Melissa
Equation in the form of a pending point y+2=1/6(×-4)
Jose Reply
write in simplest form 3 4/2
August
definition of quadratic formula
Ahmed Reply
From Google: The quadratic formula, , is used in algebra to solve quadratic equations (polynomial equations of the second degree). The general form of a quadratic equation is , where x represents a variable, and a, b, and c are constants, with . A quadratic equation has two solutions, called roots.
Melissa
what is the answer of w-2.6=7.55
What Reply
10.15
Michael
w = 10.15 You add 2.6 to both sides and then solve for w (-2.6 zeros out on the left and leaves you with w= 7.55 + 2.6)
Korin
Nataly is considering two job offers. The first job would pay her $83,000 per year. The second would pay her $66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first?
Mckenzie Reply
x > $110,000
bruce
greater than $110,000
Michael
Estelle is making 30 pounds of fruit salad from strawberries and blueberries. Strawberries cost $1.80 per pound, and blueberries cost $4.50 per pound. If Estelle wants the fruit salad to cost her $2.52 per pound, how many pounds of each berry should she use?
nawal Reply
$1.38 worth of strawberries + $1.14 worth of blueberries which= $2.52
Leitha
how
Zaione
is it right😊
Leitha
lol maybe
Robinson
8 pound of blueberries and 22 pounds of strawberries
Melissa
8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound
Melissa
8 pounds x 4.5 equal 36 22 pounds x 1.80 equal 39.60 36 + 39.60 equal 75.60 75.60 / 30 equal average 2.52 per pound
Melissa
hmmmm...... ?
Robinson
8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound
Melissa
The question asks how many pounds of each in order for her to have an average cost of $2.52. She needs 30 lb in all so 30 pounds times $2.52 equals $75.60. that's how much money she is spending on the fruit. That means she would need 8 pounds of blueberries and 22 lbs of strawberries to equal 75.60
Melissa
good
Robinson
👍
Leitha
thanks Melissa.
Leitha
nawal let's do another😊
Leitha
we can't use emojis...I see now
Leitha
Sorry for the multi post. My phone glitches.
Melissa

Get the best Elementary algebra course in your pocket!





Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask