# 8.1 Solve equations using the subtraction and addition properties  (Page 5/6)

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## Practice makes perfect

Solve Equations Using the Subtraction and Addition Properties of Equality

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

Is $y=\frac{1}{3}$ a solution of $4y+2=10y?$

yes

Is $x=\frac{3}{4}$ a solution of $5x+3=9x?$

Is $u=-\frac{1}{2}$ a solution of $8u-1=6u?$

no

Is $v=-\frac{1}{3}$ a solution of $9v-2=3v?$

In the following exercises, solve each equation.

$x+7=12$

x = 5

$y+5=-6$

$b+\frac{1}{4}=\frac{3}{4}$

$b=\frac{1}{2}$

$a+\frac{2}{5}=\frac{4}{5}$

$p+2.4=-9.3$

p = −11.7

$m+7.9=11.6$

$a-3=7$

a = 10

$m-8=-20$

$x-\frac{1}{3}=2$

$x=\frac{7}{3}$

$x-\frac{1}{5}=4$

$y-3.8=10$

y = 13.8

$y-7.2=5$

$x-15=-42$

x = −27

$z+5.2=-8.5$

$q+\frac{3}{4}=\frac{1}{2}$

$q=-\frac{1}{4}$

$p-\frac{2}{5}=\frac{2}{3}$

$y-\frac{3}{4}=\frac{3}{5}$

$y=\frac{27}{20}$

Solve Equations that Need to be Simplified

In the following exercises, solve each equation.

$c+3-10=18$

$m+6-8=15$

17

$9x+5-8x+14=20$

$6x+8-5x+16=32$

8

$-6x-11+7x-5=-16$

$-8n-17+9n-4=-41$

−20

$3\left(y-5\right)-2y=-7$

$4\left(y-2\right)-3y=-6$

2

$8\left(u+1.5\right)-7u=4.9$

$5\left(w+2.2\right)-4w=9.3$

1.7

$-5\left(y-2\right)+6y=-7+4$

$-8\left(x-1\right)+9x=-3+9$

−2

$3\left(5n-1\right)-14n+9=1-2$

$2\left(8m+3\right)-15m-4=3-5$

−4

$-\left(j+2\right)+2j-1=5$

$-\left(k+7\right)+2k+8=7$

6

$6a-5\left(a-2\right)+9=-11$

$8c-7\left(c-3\right)+4=-16$

−41

$8\left(4x+5\right)-5\left(6x\right)-x=53$

$6\left(9y-1\right)-10\left(5y\right)-3y=22$

28

Translate to an Equation and Solve

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

Five more than $x$ is equal to $21.$

The sum of $x$ and $-5$ is $33.$

x + (−5) = 33; x = 38

Ten less than $m$ is $-14.$

Three less than $y$ is $-19.$

y − 3 = −19; y = −16

The sum of $y$ and $-3$ is $40.$

Eight more than $p$ is equal to $52.$

p + 8 = 52; p = 44

The difference of $9x$ and $8x$ is $17.$

The difference of $5c$ and $4c$ is $60.$

5 c − 4 c = 60; 60

The difference of $n$ and $\frac{1}{6}$ is $\frac{1}{2}.$

The difference of $f$ and $\frac{1}{3}$ is $\frac{1}{12}.$

$f-\frac{1}{3}=\frac{1}{12};\phantom{\rule{0.2em}{0ex}}\frac{5}{12}$

The sum of $-4n$ and $5n$ is $-32.$

The sum of $-9m$ and $10m$ is $-25.$

−9 m + 10 m = −25; m = −25

Translate and Solve Applications

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

Pilar drove from home to school and then to her aunt’s house, a total of $18$ miles. The distance from Pilar’s house to school is $7$ miles. What is the distance from school to her aunt’s house?

Jeff read a total of $54$ pages in his English and Psychology textbooks. He read $41$ pages in his English textbook. How many pages did he read in his Psychology textbook?

Let p equal the number of pages read in the Psychology book 41 + p = 54. Jeff read pages in his Psychology book.

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

Eva’s daughter is $5$ years younger than her son. Eva’s son is $12$ years old. How old is her daughter?

Let d equal the daughter’s age. d = 12 − 5. Eva’s daughter’s age is 7 years old.

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

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

21 pounds

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?

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

Melissa’s math book cost $\text{22.85}$ less than her art book cost. Her math book cost $\text{93.75}.$ How much did her art book cost?

Ron’s paycheck this week was $\text{17.43}$ less than his paycheck last week. His paycheck this week was $\text{103.76}.$ How much was Ron’s paycheck last week?

$121.19 ## Everyday math Baking Kelsey needs $\frac{2}{3}$ cup of sugar for the cookie recipe she wants to make. She only has $\frac{1}{4}$ cup of sugar and will borrow the rest from her neighbor. Let $s$ equal the amount of sugar she will borrow. Solve the equation $\frac{1}{4}+s=\frac{2}{3}$ to find the amount of sugar she should ask to borrow. Construction Miguel wants to drill a hole for a $\frac{5}{\text{8}}\phantom{\rule{0.1em}{0ex}}\text{-inch}$ screw. The screw should be $\frac{1}{12}$ inch larger than the hole. Let $d$ equal the size of the hole he should drill. Solve the equation $d+\frac{1}{12}=\frac{5}{8}$ to see what size the hole should be. $d=\frac{13}{24}$ ## Writing exercises Is $-18$ a solution to the equation $3x=16-5x?$ How do you know? Write a word sentence that translates the equation $y-18=41$ and then make up an application that uses this equation in its solution. Answers will vary. ## Self check After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. If most of your checks were: …confidently. Congratulations! You have achieved the objectives 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 because topics you do not master become potholes in your road to success. In math, 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 a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. 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