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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

write in this form a/b answer should be in the simplest form 5%
August Reply
convert to decimal 9/11
August
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
Vina has $4.70 in quarters, dimes and nickels in her purse. She has eight more dimes than quarters and six more nickels than quarters. How many of each coin does she have?
Mckenzie Reply
10 quarters 16 dimes 12 nickels
Leitha
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet.
Crispy Reply
wtf. is a tail wind or headwind?
Robert
48 miles per hour with headwind and 68 miles per hour with tailwind
Leitha
average speed is 58 mph
Leitha
Into the wind (headwind), 125 mph; with wind (tailwind), 175 mph. Use time (t) = distance (d) ÷ rate (r). since t is equal both problems, then 1210/(x-25) = 1694/(×+25). solve for x gives x=150.
bruce
the jet will fly 9.68 hours to cover either distance
bruce
Riley is planning to plant a lawn in his yard. He will need 9 pounds of grass seed. He wants to mix Bermuda seed that costs $4.80 per pound with Fescue seed that costs $3.50 per pound. How much of each seed should he buy so that the overall cost will be $4.02 per pound?
Vonna Reply
33.336
Robinson
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?
Imaan Reply
Ivan has $8.75 in nickels and quarters in his desk drawer. The number of nickels is twice the number of quarters. How many coins of each type does he have?
mikayla Reply
2q=n ((2q).05) + ((q).25) = 8.75 .1q + .25q = 8.75 .35q = 8.75 q = 25 quarters 2(q) 2 (25) = 50 nickles Answer check 25 x .25 = 6.25 50 x .05 = 2.50 6.25 + 2.50 = 8.75
Melissa
John has $175 in $5 and $10 bills in his drawer. The number of $5 bills is three times the number of $10 bills. How many of each are in the drawer?
mikayla Reply
7-$10 21-$5
Robert
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? (Round your answer to the nearest tenth of a percent.)
Parker Reply
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hour longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Markeice Reply
8mph
michele
16mph
Robert
3.8 mph
Ped
16 goes into 80 5times while 20 goes into 80 4times and is 4mph faster
Robert
what is the answer for this 3×9+28÷4-8
What Reply
315
lashonna
how do you do xsquard+7x+10=0
What
(x + 2)(x + 5), then set each factor to zero and solve for x. so, x = -2 and x = -5.
bruce
I skipped it
What
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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