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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

Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles.
Dojzae Reply
LeBron needs 150 milliliters of a 30% solution of sulfuric acid for a lab experiment but only has access to a 25% and a 50% solution. How much of the 25% and how much of the 50% solution should he mix to make the 30% solution?
Xona Reply
5%
Michael
hey everyone how to do algebra
The Reply
Felecia answer 1.5 hours before he reaches her
Adriana Reply
I would like to solve the problem -6/2x
rachel Reply
12x
Andrew
how
Christian
Does the x represent a number or does it need to be graphed ?
latonya
-3/x
Venugopal
-3x is correct
Atul
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received $4,500 in interest in one year?
Stephanie Reply
Tickets for the community fair cost $12 for adults and $5 for children. On the first day of the fair, 312 tickets were sold for a total of $2204. How many adult tickets and how many child tickets were sold?
Alpha Reply
220
gayla
Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?
Tsimmuaj Reply
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
? Is there anything wrong with this passage I found the total sum for 2 jobs, but found why elaborate on extra If I total one week from the store *4 would = the month than the total is = x than x can't calculate 10 month of a year
candido
what would be wong
candido
87 divided by 3 then multiply that by 4. 116 people total.
Melissa
the actual number that has 3 out of 4 of a whole pie
candido
was having a hard time finding
Teddy
use Matrices for the 2nd question
Daniel
One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers.
Tsimmuaj Reply
26 + 37 = 63 + 8 = 71
gayla
26+37=63+8=71
ziad
11+52=63+8=71
Thisha
how do we know the answer is correct?
Thisha
23 is 11 less than 37. 23+37=63. 63+8=71. that is what the question asked for.
gayla
23 +11 = 37. 23+37=63 63+8=71
Gayla
by following the question. one number is 11 less than the other number 26+11=37 so 26+37=63+8=71
Gayla
your answer did not fit the guidelines of the question 11 is 41 less than 52.
gayla
71-8-11 =52 is this correct?
Ruel
let the number is 'x' and the other number is "x-11". if their sum is increased means: x+(x-11)+8 result will be 71. so x+(x-11)+8=71 2x-11+8=71 2x-3=71 2x=71+3 2x=74 1/2(2x=74)1/2 x=37 final answer
tesfu
just new
Muwanga
Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal?
Tsimmuaj Reply
yes math
Kenneth
company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100
gayla
need help with math to do tsi test
Toocute
me too
Christian
have you tried the TSI practice test ***tsipracticetest.com
gayla
DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players.
gustavo Reply
?
Ann
DaMarcus: 16 mi/hr Fabian: 22 mi/hr
Sherman
Joy is preparing 20 liters of a 25% saline solution. She has only a 40% solution and a 10% solution in her lab. How many liters of the 40% solution and how many liters of the 10% solution should she mix to make the 25% solution?
Wenda Reply
15 and 5
32 is 40% , & 8 is 10 % , & any 4 letters is 5%.
Karen
It felt that something is missing on the question like: 40% of what solution? 10% of what solution?
Jhea
its confusing
Sparcast
3% & 2% to complete the 25%
Sparcast
because she already has 20 liters.
Sparcast
ok I was a little confused I agree 15% & 5%
Sparcast
8,2
Karen
Jim and Debbie earned $7200. Debbie earned $1600 more than Jim earned. How much did they earned
Arleathia Reply
5600
Gloria
1600
Gloria
Bebbie: 4,400 Jim: 2,800
Jhea
A river cruise boat sailed 80 miles down the Mississippi River for 4 hours. It took 5 hours to return. Find the rate of the cruise boat in still water and the rate of the current.
Sunnyshay Reply
A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
Elizabeth Reply
ggfcc
Mike
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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