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Solve: 8 x + 9 x 5 x = −3 + 15 .

Solution

Start by combining like terms to simplify each side.

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Combine like terms. .
Divide both sides by 12 to isolate x. .
Simplify. .
Check your answer. Let x = 1
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Solve: 7 x + 6 x 4 x = −8 + 26 .

x = 2

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

n = −5

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

Solution

Simplify each side by combining like terms.

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Simplify each side. .
Divide both sides by 3 to isolate y. .
Simplify. .
Check your answer. Let y = −3
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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.

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

c = −3

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Solve: 18 22 = 12 x x 4 x .

x = 4 7

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Solve: −3 ( n 2 ) 6 = 21 .

Solution

Remember—always simplify each side first.

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Distribute. .
Simplify. .
Divide both sides by -3 to isolate n. .
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Check your answer. Let n = −7 .
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Solve: −4 ( n 2 ) 8 = 24 .

n = −6

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Solve: −6 ( n 2 ) 12 = 30 .

n = −5

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Key concepts

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

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.

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

Solve Equations That Need to be Simplified

In the following exercises, solve the equation.

8 a + 3 a 6 a = −17 + 27

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6 y 3 y + 12 y = −43 + 28

y = −1

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−9 x 9 x + 2 x = 50 2

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−5 m + 7 m 8 m = −6 + 36

m = −5

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100 16 = 4 p 10 p p

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−18 7 = 5 t 9 t 6 t

t = 5 2

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5 12 q + 1 2 q = 25 3

q = 24

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0.25 d + 0.10 d = 6 0.75

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0.05 p 0.01 p = 2 + 0.24

p = 56

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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 3 b = 18 .

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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 4 g = 24 .

6 children

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

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

$1.08

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Fuel economy Tania’s SUV gets half as many miles per gallon (mpg) as her husband’s hybrid car. The SUV gets 18 mpg . Find the miles per gallons, m , of the hybrid car, by solving the equation 1 2 m = 18 .

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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 1 3 f = 14 .

42 yards

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Writing exercises

Frida started to solve the equation −3 x = 36 by adding 3 to both sides. Explain why Frida’s method will result in the correct solution.

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Emiliano thinks x = 40 is the solution to the equation 1 2 x = 80 . Explain why he is wrong.

Answer will vary.

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Self check

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

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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.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
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)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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