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By the end of this section, you will be able to:
  • Solve equations using a general strategy
  • Classify equations

Before you get started, take this readiness quiz.

  1. Simplify: ( a 4 ) .
    If you missed this problem, review [link] .
  2. Multiply: 3 2 ( 12 x + 20 ) .
    If you missed this problem, review [link] .
  3. Simplify: 5 2 ( n + 1 ) .
    If you missed this problem, review [link] .
  4. Multiply: 3 ( 7 y + 9 ) .
    If you missed this problem, review [link] .
  5. Multiply: ( 2.5 ) ( 6.4 ) .
    If you missed this problem, review [link] .

Solve equations using the general strategy

Until now we have dealt with solving one specific form of a linear equation. It is time now to lay out one overall strategy that can be used to solve any linear equation. Some equations we solve will not require all these steps to solve, but many will.

Beginning by simplifying each side of the equation makes the remaining steps easier.

How to solve linear equations using the general strategy

Solve: −6 ( x + 3 ) = 24 .

Solution

This figure is a table that has three columns and five rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads: “Step 1. Simplify each side of the equation as much as possible.” The text in the second cell reads: “Use the Distributive Property. Notice that each side of the equation is simplified as much as possible.” The third cell contains the equation negative 6 times x plus 3, where x plus 3 is in parentheses, equals 24. Below this is the same equation with the negative 6 distributed across the parentheses: negative 6x minus 18 equals 24. In the second row of the table, the first cell says: “Step 2. Collect all variable terms on one side of the equation.” In the second cell, the instructions say: “Nothing to do—all x’s are on the left side. The third cell is blank. In the third row of the table, the first cell says: “Step 3. Collect constant terms on the other side of the equation. In the second cell, the instructions say: “To get constants only on the right, add 18 to each side. Simplify.” The third cell contains the same equation with 18 added to both sides: negative 6x minus 18 plus 18 equals 24 plus 18. Below this is the equation negative 6x equals 42. In the fourth row of the table, the first cell says: “Step 4. Make the coefficient of the variable term equal to 1.” In the second cell, the instructions say: “Divide each side by negative 6. Simplify. The third cell contains the same equation divided by negative 6 on both sides: negative 6x over negative 6 equals 42 over negative 6, with “divided by negative 6” written in red on both sides. Below this is the answer to the equation: x equals negative 7. In the fifth row of the table, the first cell says: “Step 5. Check the solution.” In the second cell, the instructions say: “Let x equal negative 7. Simplify. Multiply.” In the third cell, there is the instruction: “Check,” and to the right of this is the original equation again: negative 6 times x plus 3, with x plus 3 in parentheses, equal 24. Below this is the same equation with negative 7 substituted in for x: negative 6 times negative 7 plus 3, with negative 7 plus 3 in parentheses, might equal 24. Below this is the equation negative 6 times negative 4 might equal 24. Below this is the equation 24 equals 24, with a check mark next to it.
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Solve: 5 ( x + 3 ) = 35 .

x = 4

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Solve: 6 ( y 4 ) = −18 .

y = 1

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General strategy for solving linear equations.

  1. Simplify each side of the equation as much as possible.
    Use the Distributive Property to remove any parentheses.
    Combine like terms.
  2. Collect all the variable terms on one side of the equation.
    Use the Addition or Subtraction Property of Equality.
  3. Collect all the constant terms on the other side of the equation.
    Use the Addition or Subtraction Property of Equality.
  4. Make the coefficient of the variable term to equal to 1.
    Use the Multiplication or Division Property of Equality.
    State the solution to the equation.
  5. Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.

Solve: ( y + 9 ) = 8 .

Solution

.
Simplify each side of the equation as much as possible by distributing. .
The only y term is on the left side, so all variable terms are on the left side of the equation.
Add 9 to both sides to get all constant terms on the right side of the equation. .
Simplify. .
Rewrite y as −1 y . .
Make the coefficient of the variable term to equal to 1 by dividing both sides by −1 . .
Simplify. .
Check: .
Let y = −17 . .
.
.
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Solve: ( y + 8 ) = −2 .

y = −6

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Solve: ( z + 4 ) = −12 .

z = 8

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Solve: 5 ( a 3 ) + 5 = −10 .

Solution

.
Simplify each side of the equation as much as possible.
Distribute. .
Combine like terms. .
The only a term is on the left side, so all variable terms are on one side of the equation.
Add 10 to both sides to get all constant terms on the other side of the equation. .
Simplify. .
Make the coefficient of the variable term to equal to 1 by dividing both sides by 5 . .
Simplify. .
Check: .
Let a = 0 . .
.
.
.
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Solve: 2 ( m 4 ) + 3 = −1 .

m = 2

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Solve: 7 ( n 3 ) 8 = −15 .

n = 2

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Solve: 2 3 ( 6 m 3 ) = 8 m .

Solution

.
Distribute. .
Add m to get the variables only to the left. .
Simplify. .
Add 2 to get constants only on the right. .
Simplify. .
Divide by 5 . .
Simplify. .
Check: .
Let m = 2 . .
.
.
.
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Solve: 1 3 ( 6 u + 3 ) = 7 u .

u = 2

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Solve: 2 3 ( 9 x 12 ) = 8 + 2 x .

x = 4

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Solve: 8 2 ( 3 y + 5 ) = 0 .

Solution

.
Simplify—use the Distributive Property. .
Combine like terms. .
Add 2 to both sides to collect constants on the right. .
Simplify. .
Divide both sides by −6 . .
Simplify. .
Check: Let y = 1 3 .
.
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Solve: 12 3 ( 4 j + 3 ) = −17 .

j = 5 3

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Solve: −6 8 ( k 2 ) = −10 .

k = 5 2

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Solve: 4 ( x 1 ) 2 = 5 ( 2 x + 3 ) + 6 .

Solution

.
Distribute. .
Combine like terms. .
Subtract 4 x to get the variables only on the right side since 10 > 4 . .
Simplify. .
Subtract 21 to get the constants on left. .
Simplify. .
Divide by 6. .
Simplify. .
Check: .
Let x = 9 2 . .
.
.
.
.
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Questions & Answers

Aziza is solving this equation-2(1+x)=4x+10
Sechabe Reply
No. 3^32 -1 has exactly two divisors greater than 75 and less than 85 what is their product?
KAJAL Reply
x^2+7x-19=0 has Two solutions A and B give your answer to 3 decimal places
Adedamola Reply
please the answer to the example exercise
Patricia Reply
3. When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for20 minutes, her fitness app says she burned 278 calories. When she spent 20 minutes onthe elliptical trainer and 30 minutes circuit training she burned 473 calories. How manycalories does she burn for each minute on the elliptical trainer? How many calories doesshe burn for each minute of circuit training?
Edwin Reply
.473
Angelita
?
Angelita
John left his house in Irvine at 8:35 am to drive to a meeting in Los Angeles, 45 miles away. He arrived at the meeting at 9:50. At 3:30 pm, he left the meeting and drove home. He arrived home at 5:18.
DaYoungan Reply
p-2/3=5/6 how do I solve it with explanation pls
Adedamola Reply
P=3/2
Vanarith
1/2p2-2/3p=5p/6
James
don't understand answer
Cindy
4.5
Ruth
is y=7/5 a solution of 5y+3=10y-4
Adedamola Reply
yes
James
don't understand answer
Cindy
Lucinda has a pocketful of dimes and quarters with a value of $6.20. The number of dimes is 18 more than 3 times the number of quarters. How many dimes and how many quarters does Lucinda have?
Rhonda Reply
Find an equation for the line that passes through the point P ( 0 , − 4 ) and has a slope 8/9 .
Gabriel Reply
is that a negative 4 or positive 4?
Felix
y = mx + b
Felix
if negative -4, then -4=8/9(0) + b
Felix
-4=b
Felix
if positive 4, then 4=b
Felix
then plug in y=8/9x - 4 or y=8/9x+4
Felix
Macario is making 12 pounds of nut mixture with macadamia nuts and almonds. macadamia nuts cost $9 per pound and almonds cost $5.25 per pound. how many pounds of macadamia nuts and how many pounds of almonds should macario use for the mixture to cost $6.50 per pound to make?
Cherry Reply
Nga and Lauren bought a chest at a flea market for $50. They re-finished it and then added a 350 % mark - up
Makaila Reply
$1750
Cindy
the sum of two Numbers is 19 and their difference is 15
Abdulai Reply
2, 17
Jose
interesting
saw
4,2
Cindy
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 13 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
Rafi Reply
hola saben como aser un valor de la expresión
NAILEA
integer greater than 2 and less than 12
Emily Reply
2 < x < 12
Felix
I'm guessing you are doing inequalities...
Felix
Actually, translating words into algebraic expressions / equations...
Felix
hi
Darianna
hello
Mister
Eric here
Eric
6
Cindy
Practice Key Terms 3

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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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