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Verify a Solution of an Equation

In the following exercises, determine whether the given value is a solution to the equation.

Is y = 5 3 a solution of
6 y + 10 = 12 y ?

yes

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Is x = 9 4 a solution of
4 x + 9 = 8 x ?

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Is u = 1 2 a solution of
8 u 1 = 6 u ?

no

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Is v = 1 3 a solution of
9 v 2 = 3 v ?

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Solve Equations using the Subtraction and Addition Properties of Equality

In the following exercises, solve each equation using the Subtraction and Addition Properties of Equality.

y + 45 = −66

y = −111

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p + 2.4 = −9.3

p = −11.7

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m 18 = −200

m = −182

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x 165 = −420

x = −255

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z + 0.52 = −8.5

z = −9.02

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q + 3 4 = 1 2

q = 1 4

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p 2 5 = 2 3

p = 16 15

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Solve Equations that Require Simplification

In the following exercises, solve each equation.

c + 31 10 = 46

c = 25

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9 x + 5 8 x + 14 = 20

x = 1

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6 x + 8 5 x + 16 = 32

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−6 x 11 + 7 x 5 = −16

x = 0

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−8 n 17 + 9 n 4 = −41

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5 ( y 6 ) 4 y = −6

y = 8 y = 24

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9 ( y 2 ) 8 y = −16

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8 ( u + 1.5 ) 7 u = 4.9

u = −7.1

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5 ( w + 2.2 ) 4 w = 9.3

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6 a 5 ( a 2 ) + 9 = −11

a = −30

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8 c 7 ( c 3 ) + 4 = −16

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6 ( y 2 ) 5 y = 4 ( y + 3 )
4 ( y 1 )

y = 28

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9 ( x 1 ) 8 x = −3 ( x + 5 )
+ 3 ( x 5 )

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3 ( 5 n 1 ) 14 n + 9
= 10 ( n 4 ) 6 n 4 ( n + 1 )

n = −50

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2 ( 8 m + 3 ) 15 m 4
= 9 ( m + 6 ) 2 ( m 1 ) 7 m

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( j + 2 ) + 2 j 1 = 5

j = 8

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( k + 7 ) + 2 k + 8 = 7

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( 1 4 a 3 4 ) + 5 4 a = −2

a = 11 4

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( 2 3 d 1 3 ) + 5 3 d = −4

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8 ( 4 x + 5 ) 5 ( 6 x ) x
= 53 6 ( x + 1 ) + 3 ( 2 x + 2 )

x = 13

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6 ( 9 y 1 ) 10 ( 5 y ) 3 y
= 22 4 ( 2 y 12 ) + 8 ( y 6 )

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Translate to an Equation and Solve

In the following exercises, translate to an equation and then solve it.

Nine more than x is equal to 52.

x + 9 = 52 ; x = 43

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The sum of x and −15 is 23.

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Ten less than m is −14 .

m 10 = −14 ; m = −4

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Three less than y is −19 .

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The sum of y and −30 is 40.

y + ( −30 ) = 40 ; y = 70

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Twelve more than p is equal to 67.

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The difference of 9 x and 8 x is 107.

9 x 8 x = 107 ; 107

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The difference of 5 c and 4 c is 602.

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The difference of n and 1 6 is 1 2 .

n 1 6 = 1 2 ; 2 3

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The difference of f and 1 3 is 1 12 .

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The sum of −4 n and 5 n is −82 .

−4 n + 5 n = −82 ; 82

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The sum of −9 m and 10 m is −95 .

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Translate and Solve Applications

In the following exercises, translate into an equation and solve.

Distance Avril rode her bike a total of 18 miles, from home to the library and then to the beach. The distance from Avril’s house to the library is 7 miles. What is the distance from the library to the beach?

11 miles

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Reading Jeff read a total of 54 pages in his History and Sociology textbooks. He read 41 pages in his History textbook. How many pages did he read in his Sociology textbook?

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Age Eva’s daughter is 15 years younger than her son. Eva’s son is 22 years old. How old is her daughter?

7 years old

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Age Pablo’s father is 3 years older than his mother. Pablo’s mother is 42 years old. How old is his father?

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Groceries For a family birthday dinner, Celeste bought a turkey that weighed 5 pounds less than the one she bought for Thanksgiving. The birthday turkey weighed 16 pounds. How much did the Thanksgiving turkey weigh?

21 pounds

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Weight Allie weighs 8 pounds less than her twin sister Lorrie. Allie weighs 124 pounds. How much does Lorrie weigh?

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Health Connor’s temperature was 0.7 degrees higher this morning than it had been last night. His temperature this morning was 101.2 degrees. What was his temperature last night?

100.5 degrees

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Health The nurse reported that Tricia’s daughter had gained 4.2 pounds since her last checkup and now weighs 31.6 pounds. How much did Tricia’s daughter weigh at her last checkup?

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Salary Ron’s paycheck this week was $17.43 less than his paycheck last week. His paycheck this week was $103.76. How much was Ron’s paycheck last week?

$121.19

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Textbooks Melissa’s math book cost $22.85 less than her art book cost. Her math book cost $93.75. How much did her art book cost?

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Everyday math

Construction Miguel wants to drill a hole for a 5 8 inch screw. The hole should be 1 12 inch smaller than the screw. Let d equal the size of the hole he should drill. Solve the equation d 1 12 = 5 8 to see what size the hole should be.

d = 17 24 inch

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Baking Kelsey needs 2 3 cup of sugar for the cookie recipe she wants to make. She only has 3 8 cup of sugar and will borrow the rest from her neighbor. Let s equal the amount of sugar she will borrow. Solve the equation 3 8 + s = 2 3 to find the amount of sugar she should ask to borrow.

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Writing exercises

Is −8 a solution to the equation 3 x = 16 5 x ? How do you know?

No. Justifications will vary.

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What is the first step in your solution to the equation 10 x + 2 = 4 x + 26 ?

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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 six 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 “verify a solution of an equation,” “solve equations using the subtraction and addition properties of equality,” “solve equations that require simplification,” “translate to an equation and solve,” and “translate and solve applications.” The rest of the cells are blank.

If most of your checks were:

…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential - every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.

Questions & Answers

Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles.
Dojzae Reply
LeBron needs 150 milliliters of a 30% solution of sulfuric acid for a lab experiment but only has access to a 25% and a 50% solution. How much of the 25% and how much of the 50% solution should he mix to make the 30% solution?
Xona Reply
5%
Michael
hey everyone how to do algebra
The Reply
Felecia answer 1.5 hours before he reaches her
Adriana Reply
I would like to solve the problem -6/2x
rachel Reply
12x
Andrew
how
Christian
Does the x represent a number or does it need to be graphed ?
latonya
-3/x
Venugopal
-3x is correct
Atul
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received $4,500 in interest in one year?
Stephanie Reply
Tickets for the community fair cost $12 for adults and $5 for children. On the first day of the fair, 312 tickets were sold for a total of $2204. How many adult tickets and how many child tickets were sold?
Alpha Reply
220
gayla
Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?
Tsimmuaj Reply
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
? Is there anything wrong with this passage I found the total sum for 2 jobs, but found why elaborate on extra If I total one week from the store *4 would = the month than the total is = x than x can't calculate 10 month of a year
candido
what would be wong
candido
87 divided by 3 then multiply that by 4. 116 people total.
Melissa
the actual number that has 3 out of 4 of a whole pie
candido
was having a hard time finding
Teddy
use Matrices for the 2nd question
Daniel
One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers.
Tsimmuaj Reply
26 + 37 = 63 + 8 = 71
gayla
26+37=63+8=71
ziad
11+52=63+8=71
Thisha
how do we know the answer is correct?
Thisha
23 is 11 less than 37. 23+37=63. 63+8=71. that is what the question asked for.
gayla
23 +11 = 37. 23+37=63 63+8=71
Gayla
by following the question. one number is 11 less than the other number 26+11=37 so 26+37=63+8=71
Gayla
your answer did not fit the guidelines of the question 11 is 41 less than 52.
gayla
71-8-11 =52 is this correct?
Ruel
let the number is 'x' and the other number is "x-11". if their sum is increased means: x+(x-11)+8 result will be 71. so x+(x-11)+8=71 2x-11+8=71 2x-3=71 2x=71+3 2x=74 1/2(2x=74)1/2 x=37 final answer
tesfu
just new
Muwanga
Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal?
Tsimmuaj Reply
yes math
Kenneth
company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100
gayla
need help with math to do tsi test
Toocute
me too
Christian
have you tried the TSI practice test ***tsipracticetest.com
gayla
DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players.
gustavo Reply
?
Ann
DaMarcus: 16 mi/hr Fabian: 22 mi/hr
Sherman
Joy is preparing 20 liters of a 25% saline solution. She has only a 40% solution and a 10% solution in her lab. How many liters of the 40% solution and how many liters of the 10% solution should she mix to make the 25% solution?
Wenda Reply
15 and 5
32 is 40% , & 8 is 10 % , & any 4 letters is 5%.
Karen
It felt that something is missing on the question like: 40% of what solution? 10% of what solution?
Jhea
its confusing
Sparcast
3% & 2% to complete the 25%
Sparcast
because she already has 20 liters.
Sparcast
ok I was a little confused I agree 15% & 5%
Sparcast
8,2
Karen
Jim and Debbie earned $7200. Debbie earned $1600 more than Jim earned. How much did they earned
Arleathia Reply
5600
Gloria
1600
Gloria
Bebbie: 4,400 Jim: 2,800
Jhea
A river cruise boat sailed 80 miles down the Mississippi River for 4 hours. It took 5 hours to return. Find the rate of the cruise boat in still water and the rate of the current.
Sunnyshay Reply
A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
Elizabeth Reply
ggfcc
Mike
Practice Key Terms 1

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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