# 8.2 Solve equations using the division and multiplication properties  (Page 2/2)

 Page 2 / 2

Solve: $8x+9x-5x=-3+15.$

## Solution

Start by combining like terms to simplify each side. Combine like terms. Divide both sides by 12 to isolate x. Simplify. Check your answer. Let $x=1$    Solve: $7x+6x-4x=-8+26.$

x = 2

Solve: $11n-3n-6n=7-17.$

n = −5

Solve: $11-20=17y-8y-6y.$

## Solution

Simplify each side by combining like terms. Simplify each side. Divide both sides by 3 to isolate y. Simplify. Check your answer. Let $y=-3$    Notice that the variable ended up on the right side of the equal sign when we solved the equation. You may prefer to take one more step to write the solution with the variable on the left side of the equal sign.

Solve: $18-27=15c-9c-3c.$

c = −3

Solve: $18-22=12x-x-4x.$

$x=-\frac{4}{7}$

Solve: $-3\left(n-2\right)-6=21.$

## Solution

Remember—always simplify each side first. Distribute. Simplify. Divide both sides by -3 to isolate n.  Check your answer. Let $n=-7$ .     Solve: $-4\left(n-2\right)-8=24.$

n = −6

Solve: $-6\left(n-2\right)-12=30.$

n = −5

## Key concepts

• Division and Multiplication Properties of Equality
• Division Property of Equality: For all real numbers a, b, c, and $c\ne 0$ , if $a=b$ , then $ac=bc$ .
• Multiplication Property of Equality: For all real numbers a, b, c, if $a=b$ , then $ac=bc$ .

## Practice makes perfect

Solve Equations Using the Division and Multiplication Properties of Equality

In the following exercises, solve each equation for the variable using the Division Property of Equality and check the solution.

$8x=32$

$7p=63$

9

$-5c=55$

$-9x=-27$

3

$-90=6y$

$-72=12y$

−7

$-16p=-64$

$-8m=-56$

7

$0.25z=3.25$

$0.75a=11.25$

15

$-3x=0$

$4x=0$

0

In the following exercises, solve each equation for the variable using the Multiplication Property of Equality and check the solution.

$\frac{x}{4}=15$

$\frac{z}{2}=14$

28

$-20=\frac{\phantom{\rule{0.4em}{0ex}}q}{-5}$

$\frac{\phantom{\rule{0.4em}{0ex}}c}{-3}=-12$

36

$\frac{y}{9}=-6$

$\frac{q}{6}=-8$

−48

$\frac{m}{-12}=5$

$-4=\frac{p}{-20}$

80

$\frac{2}{3}\phantom{\rule{0.1em}{0ex}}y=18$

$\frac{3}{5}\phantom{\rule{0.1em}{0ex}}r=15$

25

$-\frac{5}{8}\phantom{\rule{0.1em}{0ex}}w=40$

$24=-\frac{3}{4}\phantom{\rule{0.1em}{0ex}}x$

−32

$-\frac{2}{5}=\frac{1}{10}\phantom{\rule{0.1em}{0ex}}a$

$-\frac{1}{3}\phantom{\rule{0.1em}{0ex}}q=-\frac{5}{6}$

5/2

Solve Equations That Need to be Simplified

In the following exercises, solve the equation.

$8a+3a-6a=-17+27$

$6y-3y+12y=-43+28$

y = −1

$-9x-9x+2x=50-2$

$-5m+7m-8m=-6+36$

m = −5

$100-16=4p-10p-p$

$-18-7=5t-9t-6t$

$t=\frac{5}{2}$

$\frac{7}{8}\phantom{\rule{0.1em}{0ex}}n-\frac{3}{4}\phantom{\rule{0.1em}{0ex}}n=9+2$

$\frac{5}{12}\phantom{\rule{0.1em}{0ex}}q+\frac{1}{2}\phantom{\rule{0.1em}{0ex}}q=25-3$

q = 24

$0.25d+0.10d=6-0.75$

$0.05p-0.01p=2+0.24$

p = 56

## Everyday math

Balloons Ramona bought $18$ balloons for a party. She wants to make $3$ equal bunches. Find the number of balloons in each bunch, $b,$ by solving the equation $3b=18.$

Teaching Connie’s kindergarten class has $24$ children. She wants them to get into $4$ equal groups. Find the number of children in each group, $g,$ by solving the equation $4g=24.$

6 children

Ticket price Daria paid $\text{36.25}$ for $5$ children’s tickets at the ice skating rink. Find the price of each ticket, $p,$ by solving the equation $5p=36.25.$

Unit price Nishant paid $\text{12.96}$ for a pack of $12$ juice bottles. Find the price of each bottle, $b,$ by solving the equation $12b=12.96.$

$1.08 Fuel economy Tania’s SUV gets half as many miles per gallon (mpg) as her husband’s hybrid car. The SUV gets $\text{18 mpg}.$ Find the miles per gallons, $m,$ of the hybrid car, by solving the equation $\frac{1}{2}\phantom{\rule{0.1em}{0ex}}m=18.$ Fabric The drill team used $14$ yards of fabric to make flags for one-third of the members. Find how much fabric, $f,$ they would need to make flags for the whole team by solving the equation $\frac{1}{3}\phantom{\rule{0.1em}{0ex}}f=14.$ 42 yards ## Writing exercises Frida started to solve the equation $-3x=36$ by adding $3$ to both sides. Explain why Frida’s method will result in the correct solution. Emiliano thinks $x=40$ is the solution to the equation $\frac{1}{2}\phantom{\rule{0.1em}{0ex}}x=80.$ Explain why he is wrong. Answer will vary. ## Self check After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. After reviewing this checklist, what will you do to become confident for all objectives? #### Questions & Answers explain and give four Example hyperbolic function Lukman Reply The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu SABAL Reply 1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3 Pawel 2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31 Pawel Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113? mariel Reply Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113 Pawel how do I set up the problem? Harshika Reply what is a solution set? Harshika find the subring of gaussian integers? Rofiqul hello, I am happy to help! Shirley Reply please can go further on polynomials quadratic Abdullahi hi mam Mark I need quadratic equation link to Alpa Beta Abdullahi Reply find the value of 2x=32 Felix Reply divide by 2 on each side of the equal sign to solve for x corri X=16 Michael Want to review on complex number 1.What are complex number 2.How to solve complex number problems. Beyan yes i wantt to review Mark use the y -intercept and slope to sketch the graph of the equation y=6x Only Reply how do we prove the quadratic formular Seidu Reply please help me prove quadratic formula Darius hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher Shirley Reply thank you help me with how to prove the quadratic equation Seidu may God blessed u for that. Please I want u to help me in sets. Opoku what is math number Tric Reply 4 Trista x-2y+3z=-3 2x-y+z=7 -x+3y-z=6 Sidiki Reply can you teacch how to solve that🙏 Mark Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411 Brenna (61/11,41/11,−4/11) Brenna x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11 Brenna Need help solving this problem (2/7)^-2 Simone Reply x+2y-z=7 Sidiki what is the coefficient of -4× Mehri Reply -1 Shedrak the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1 Alfred Reply A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place. Kimberly Reply Jeannette has$5 and \$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
What is the expressiin for seven less than four times the number of nickels
How do i figure this problem out.
how do you translate this in Algebraic Expressions
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight? By OpenStax By OpenStax By JavaChamp Team By Jessica Collett By Mariah Hauptman By Dionne Mahaffey By Lakeima Roberts By OpenStax By OpenStax By Madison Christian