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Mathematics LibreTexts

2.2: Use a General Strategy to Solve Linear Equations

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Summary

By the end of this section, you will be able to:

  • Use the commutative and associative properties
  • Use the properties of identity, inverse, and zero
  • Simplify expressions using the Distributive Property

Before you get started, take this readiness quiz.

  1. Simplify: 32(12x+20).
  2. Simplify: 52(n+1).
  3. Find the LCD of 56 and 14.

Solve Linear Equations Using a General Strategy

Solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that makes it a true statement. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle!

Definition: SOLUTION OF AN EQUATION

A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

To determine whether a number is a solution to an equation, we substitute the value for the variable in the equation. If the resulting equation is a true statement, then the number is a solution of the equation.

how-to.png DETERMINE WHETHER A NUMBER IS A SOLUTION TO AN EQUATION

  1. Substitute the number for the variable in the equation.
  2. Simplify the expressions on both sides of the equation.
  3. Determine whether the resulting equation is true.
    • If it is true, the number is a solution.
    • If it is not true, the number is not a solution.

Example 2.2.1

Determine whether the values are solutions to the equation: 5y+3=10y4.

  1. y=35
  2. y=75
Solution

Since a solution to an equation is a value of the variable that makes the equation true, begin by substituting the value of the solution for the variable.

a.

  5y+3=10y4
Substitute 35 for y 5(35)+3?=10(35)4
Multiply. 3+3?=64
Simplify. 62

Since y=35 does not result in a true equation, y=35 is not a solution to the equation 5y+3=10y4.

b.

  5y+3=10y4
Substitute 75 for y 5(75)+3?=10(75)4
Multiply. 7+3?=144
Simplify. 10=10

Since y=75 results in a true equation, y=75 is a solution to the equation 5y+3=10y4.

try-it.png 2.2.1A

Determine whether the values are solutions to the equation: 9y+2=6y+3.

  1. y=43
  2. y=13
Answer a

 no 

Answer b

yes

try-it.png 2.2.1B

Determine whether the values are solutions to the equation: 4x2=2x+1.

  1. x=32
  2. x=12
Answer a

yes

Answer b

no

There are many types of equations that we will learn to solve. In this section we will focus on a linear equation.

Definition: LINEAR EQUATION

A linear equation is an equation in one variable that can be written, where a and b are real numbers and a0, as:

ax+b=0

To solve a linear equation it is a good idea to have an overall strategy that can be used to solve any linear equation. In the next example, we will give the steps of a general strategy for solving any linear equation. Simplifying each side of the equation as much as possible first makes the rest of the steps easier.

EXAMPLE 2.2.2

Solve: 7(n3)8=15

Solution
CNX_IntAlg_Figure_02_01_003a_img.jpg
Figure_02_01_003b_img.jpg" />
CNX_IntAlg_Figure_02_01_003c_img.jpg
Figure_02_01_003d_img.jpg" />

CNX_IntAlg_Figure_02_01_003e_img.jpg

try-it.png 2.2.2A

Solve: 2(m4)+3=1.

Answer

m=2

try-it.png 2.2.2B

Solve: 5(a3)+5=10.

Answer

a=0

These steps are summarized in the General Strategy for Solving Linear Equations below.

how-to.png SOLVE LINEAR EQUATIONS USING A GENERAL STRATEGY

  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 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.

EXAMPLE 2.2.3

Solve: 23(3m6)=5m.

Answer
  23(3m6)=5m
Distribute. 2m4=5m
Add m to both sides to get the variables only on the left. CNX_IntAlg_Figure_02_01_004h_img.jpg
Simplify. 3m4=5
Add 4 to both sides to get constants only on the right. CNX_IntAlg_Figure_02_01_004j_img.jpg
Simplify. 3m=9
Divide both sides by three. CNX_IntAlg_Figure_02_01_004l_img.jpg
Simplify. m=3

 

Check: CNX_IntAlg_Figure_02_01_004f_img.jpg
Let m=3 CNX_IntAlg_Figure_02_01_004b_img.jpg
  CNX_IntAlg_Figure_02_01_004c_img.jpg
  CNX_IntAlg_Figure_02_01_004d_img.jpg
  CNX_IntAlg_Figure_02_01_004e_img.jpg

try-it.png 2.2.3A

Solve: 13(6u+3)=7u.

Answer

u=2

try-it.png 2.2.3B

Solve: 23(9x12)=8+2x.

Answer

x=4

We can solve equations by getting all the variable terms to either side of the equal sign. By collecting the variable terms on the side where the coefficient of the variable is larger, we avoid working with some negatives. This will be a good strategy when we solve inequalities later in this chapter. It also helps us prevent errors with negatives.

EXAMPLE 2.2.4

Solve: 4(x1)2=5(2x+3)+6.

Answer
      4(x1)2=5(2x+3)+6
Distribute.    

4x42=10x+15+6

Combine like terms.     4x6=10x+21
Subtract 4x from each side to get the variables only on the right since 10>4.     CNX_IntAlg_Figure_02_01_005j_img.jpg
Simplify.     6=6x+21
Subtract 21 from each side to get the constants on left.     CNX_IntAlg_Figure_02_01_005l_img.jpg
Simplify.     27=6x
Divide both sides by 6.     CNX_IntAlg_Figure_02_01_005n_img.jpg
Simplify.     92=x
Check: CNX_IntAlg_Figure_02_01_005g_img.jpg      
Let x=92. CNX_IntAlg_Figure_02_01_005b_img.jpg      
  CNX_IntAlg_Figure_02_01_005c_img.jpg    
  CNX_IntAlg_Figure_02_01_005d_img.jpg      
  CNX_IntAlg_Figure_02_01_005e_img.jpg      
  CNX_IntAlg_Figure_02_01_005f_img.jpg      

try-it.png 2.2.4A

Solve: 6(p3)7=5(4p+3)12.

Answer

p=2

try-it.png 2.2.4B

Solve: 8(q+1)5=3(2q4)1.

Answer

q=8

EXAMPLE 2.2.5

Solve: 10[38(2s5)]=15(405s).

Answer
  10[38(2s5)]=15(405s)
Simplify from the innermost parentheses first. 10[316s+40]=15(405s)
Combine like terms in the brackets. 10[4316s]=15(405s)
Distribute. 430160s=60075s
Add 160s to both sides to get the variable to the right. CNX_IntAlg_Figure_02_01_006l_img.jpg
Simplify. 430=600+85s
Subtract 600 from both sides to get the constant to the left. CNX_IntAlg_Figure_02_01_006n_img.jpg
Simplify. 170=85s
Divide both sides by 85. CNX_IntAlg_Figure_02_01_006p_img.jpg
Simplify. \(\-2=s)
Check: 10[38(2s5)]=15(405s)  
Let s=2. CNX_IntAlg_Figure_02_01_006b_img.jpg  
  CNX_IntAlg_Figure_02_01_006c_img.jpg  
  CNX_IntAlg_Figure_02_01_006d_img.jpg  
  CNX_IntAlg_Figure_02_01_006e_img.jpg  
  CNX_IntAlg_Figure_02_01_006f_img.jpg  
  CNX_IntAlg_Figure_02_01_006g_img.jpg  

try-it.png 2.2.5A

Solve: 6[42(7y1)]=8(138y).

Answer

y=175

try-it.png 2.2.5B

Solve: 12[15(4z1)]=3(24+11z).

Answer

z=0

Classify Equations

Whether or not an equation is true depends on the value of the variable. The equation 7x+8=13 is true when we replace the variable, x, with the value 3, but not true when we replace x with any other value. An equation like this is called a conditional equation. All the equations we have solved so far are conditional equations.

Definition: CONDITIONAL EQUATION

An equation that is true for one or more values of the variable and false for all other values of the variable is a conditional equation.

Now let’s consider the equation 7y+14=7(y+2). Do you recognize that the left side and the right side are equivalent? Let’s see what happens when we solve for y.

Solve:

  7y+14=7(y+2)
Distribute. 7y+14=7y+14
Subtract 7y to each side to get the y's to one side. 7y7y+14=7y7y+14
Simplify—the y’s are eliminated. 14=14
  But 14=14 is true.

This means that the equation 7y+14=7(y+2) is true for any value of y. We say the solution to the equation is all of the real numbers. An equation that is true for any value of the variable is called an identity.

Definition: IDENTITY

An equation that is true for any value of the variable is called an identity.

The solution of an identity is valid for all real numbers.

What happens when we solve the equation 8z=8z+9?

Solve:

  8z=8z+9
Add 8z to both sides to leave the constant alone on the right. 8z+8z=8z+8z+9
Simplify—the z's are eliminated. 09
  But 09.

Solving the equation 8z=8z+9 led to the false statement 0=9. The equation 8z=8z+9 will not be true for any value of z. It has no solution. An equation that has no solution, or that is false for all values of the variable, is called a contradiction.

definition: CONTRADICTION

An equation that is false for all values of the variable is called a contradiction.

A contradiction has no solution.

The next few examples will ask us to classify an equation as conditional, an identity, or as a contradiction.

EXAMPLE 2.2.6

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: 6(2n1)+3=2n8+5(2n+1).

Answer
  6(2n1)+3=2n8+5(2n+1)
Distribute. 12n6+3=2n8+10n+5
Combine like terms. 12n3=12n3
Subtract 12n from each side to get the n’s to one side. CNX_IntAlg_Figure_02_01_009d_img.jpg
Simplify. 3=3
This is a true statement. The equation is an identity.
  The solution is all real numbers.

try-it.png 2.2.6A

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: 4+9(3x7)=42x13+23(3x2).

Answer

identity; all real numbers

Exercise 2.2.6B

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: 8(13x)+15(2x+7)=2(x+50)+4(x+3)+1.

Answer

identity; all real numbers

EXAMPLE 2.2.7

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: 8+3(a4)=0.

Answer
  CNX_IntAlg_Figure_02_01_010a_img.jpg
Distribute. CNX_IntAlg_Figure_02_01_010b_img.jpg
Combine like terms. CNX_IntAlg_Figure_02_01_010c_img.jpg
Add 4 to both sides. CNX_IntAlg_Figure_02_01_010d_img.jpg
Simplify. CNX_IntAlg_Figure_02_01_010e_img.jpg
Divide. CNX_IntAlg_Figure_02_01_010f_img.jpg
Simplify. CNX_IntAlg_Figure_02_01_010g_img.jpg
The equation is true when a=43. This is a conditional equation.
 

The solution is a=43.

Exercise 2.2.7A

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: 11(q+3)5=19.

Answer

conditional equation; q=911

Exercise 2.2.7B

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: 6+14(k8)=95.

Answer

conditional equation; k=20114

EXAMPLE 2.2.8

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: 5m+3(9+3m)=2(7m11).

Answer
  CNX_IntAlg_Figure_02_01_011a_img.jpg
Distribute. CNX_IntAlg_Figure_02_01_011b_img.jpg
Combine like terms. CNX_IntAlg_Figure_02_01_011c_img.jpg
Subtract 14m from both sides. CNX_IntAlg_Figure_02_01_011d_img.jpg
Simplify. CNX_IntAlg_Figure_02_01_011e_img.jpg
But 2722. The equation is a contradiction.
  It has no solution.

Exercise 2.2.8A

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: 12c+5(5+3c)=3(9c4).

Answer

contradiction; no solution

Exercise 2.2.8B

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution:4(7d+18)=13(3d2)11d.

Answer

contradiction; no solution

We summarize the methods for classifying equations in the table.

Type of equation What happens when you solve it? Solution
Conditional Equation True for one or more values of the variables and false for all other values One or more values
Identity True for any value of the variable All real numbers
Contradiction False for all values of the variable No solution
 

Solve Equations with Fraction or Decimal Coefficients

We could use the General Strategy to solve the next example. This method would work fine, but many students do not feel very confident when they see all those fractions. So, we are going to show an alternate method to solve equations with fractions. This alternate method eliminates the fractions.

We will apply the Multiplication Property of Equality and multiply both sides of an equation by the least common denominator (LCD) of all the fractions in the equation. The result of this operation will be a new equation, equivalent to the first, but without fractions. This process is called clearing the equation of fractions.

To clear an equation of decimals, we think of all the decimals in their fraction form and then find the LCD of those denominators.

EXAMPLE 2.2.9: How to Solve Equations with Fraction or Decimal Coefficients

Solve: 112x+56=34.

Answer
CNX_IntAlg_Figure_02_01_012a_img.jpg

CNX_IntAlg_Figure_02_01_012c_img.jpg

Exercise 2.2.9A

Solve: 14x+12=58.

Answer

x=12

Exercise 2.2.9B

Solve: 18x+12=14.

Answer

x=2

Notice in the previous example, once we cleared the equation of fractions, the equation was like those we solved earlier in this chapter. We changed the problem to one we already knew how to solve. We then used the General Strategy for Solving Linear Equations.

how-to.png SOLVE EQUATIONS WITH FRACTION OR DECIMAL COEFFICIENTS.

  1. Find the least common denominator (LCD) of all the fractions and decimals (in fraction form) in the equation.
  2. Multiply both sides of the equation by that LCD. This clears the fractions and decimals.
  3. Solve using the General Strategy for Solving Linear Equations.

EXAMPLE 2.2.10

Solve: 5=12y+23y34y.

Answer

We want to clear the fractions by multiplying both sides of the equation by the LCD of all the fractions in the equation.

Find the LCD of all fractions in the equation. CNX_IntAlg_Figure_02_01_013e_img.jpg
The LCD is 12.  
Multiply both sides of the equation by 12. CNX_IntAlg_Figure_02_01_013f_img.jpg
Distribute. CNX_IntAlg_Figure_02_01_013g_img.jpg
Simplify—notice, no more fractions. CNX_IntAlg_Figure_02_01_013h_img.jpg
Combine like terms. CNX_IntAlg_Figure_02_01_013i_img (1).jpg
Divide by five. CNX_IntAlg_Figure_02_01_013j_img.jpg
Simplify. CNX_IntAlg_Figure_02_01_013k_img.jpg
Check: CNX_IntAlg_Figure_02_01_013a_img.jpg  
Let y=12. CNX_IntAlg_Figure_02_01_013b_img.jpg  
  CNX_IntAlg_Figure_02_01_013c_img.jpg  
  CNX_IntAlg_Figure_02_01_013d_img.jpg  

Exercise 2.2.10A

Solve: 7=12x+34x23x.

Answer

x=12

Exercise 2.2.10B

Solve: 1=12u+14u23u.

Answer

u=12

In the next example, we’ll distribute before we clear the fractions.

EXAMPLE 2.2.11

Solve: 12(y5)=14(y1).

Answer
  CNX_IntAlg_Figure_02_01_014c_img.jpg
Distribute. CNX_IntAlg_Figure_02_01_014d_img.jpg
Simplify. CNX_IntAlg_Figure_02_01_014e_img.jpg
Multiply by the LCD, four. CNX_IntAlg_Figure_02_01_014f_img.jpg
Distribute. CNX_IntAlg_Figure_02_01_014g_img.jpg
Simplify. CNX_IntAlg_Figure_02_01_014h_img.jpg
Collect the variables to the left. CNX_IntAlg_Figure_02_01_014i_img.jpg
Simplify. CNX_IntAlg_Figure_02_01_014j_img.jpg
Collect the constants to the right. CNX_IntAlg_Figure_02_01_014k_img.jpg
Simplify. CNX_IntAlg_Figure_02_01_014l_img.jpg
An alternate way to solve this equation is to clear the fractions without distributing first. If you multiply the factors correctly, this method will be easier.
  CNX_IntAlg_Figure_02_01_014m_img.jpg
Multiply by the LCD, 4. CNX_IntAlg_Figure_02_01_014n_img.jpg
Multiply four times the fractions. CNX_IntAlg_Figure_02_01_014o_img.jpg
Distribute. CNX_IntAlg_Figure_02_01_014p_img.jpg
Collect the variables to the left. CNX_IntAlg_Figure_02_01_014q_img (1).jpg
Simplify. CNX_IntAlg_Figure_02_01_014r_img.jpg
Collect the constants to the right. CNX_IntAlg_Figure_02_01_014s_img.jpg
Simplify. CNX_IntAlg_Figure_02_01_014t_img.jpg
Check: CNX_IntAlg_Figure_02_01_014a_img.jpg  
Let y=9. CNX_IntAlg_Figure_02_01_014b_img.jpg  
Finish the check on your own.

Exercise 2.2.11A

Solve: 15(n+3)=14(n+2).

Answer

n=2

Exercise 2.2.11B

Solve: 12(m3)=14(m7).

Answer

m=1

When you multiply both sides of an equation by the LCD of the fractions, make sure you multiply each term by the LCD—even if it does not contain a fraction.

EXAMPLE 2.2.12

Solve: 4q+32+6=3q+54

Answer
  4q+32+6=3q+54
Multiply both sides by the LCD, 4. CNX_IntAlg_Figure_02_01_015d_img.jpg
Distribute. CNX_IntAlg_Figure_02_01_015e_img.jpg
Simplify. \(\2(4 q+3)+24=3 q+5)
  8q+6+24=3q+5
  8q+30=3q+5
Collect the variables to the left. CNX_IntAlg_Figure_02_01_015i_img.jpg
Simplify. 5q+30=5
Collect the constants to the right. CNX_IntAlg_Figure_02_01_015k_img.jpg
Simplify. 5q=25
Divide both sides by five. CNX_IntAlg_Figure_02_01_015m_img.jpg
Simplify. q=5
Check: 4q+32+6=3q+54  
Let q=5. CNX_IntAlg_Figure_02_01_015b_img.jpg  
Finish the check on your own.  

Exercise 2.2.12A

Solve: 3r+56+1=4r+33.

Answer

r=3

Exercise 2.2.12B

Solve: 2s+32+1=3s+24.

Answer

s=8

Some equations have decimals in them. This kind of equation may occur when we solve problems dealing with money or percentages. But decimals can also be expressed as fractions. For example, 0.7=710 and 0.29=29100. So, with an equation with decimals, we can use the same method we used to clear fractions—multiply both sides of the equation by the least common denominator.

The next example uses an equation that is typical of the ones we will see in the money applications in a later section. Notice that we will clear all decimals by multiplying by the LCD of their fraction form.

EXAMPLE 2.2.13

Solve: 0.25x+0.05(x+3)=2.85.

Answer

Look at the decimals and think of the equivalent fractions:

0.25=25100,0.05=5100,2.85=285100.

Notice, the LCD is 100. By multiplying by the LCD we will clear the decimals from the equation.

  CNX_IntAlg_Figure_02_01_016a_img.jpg
Distribute first. CNX_IntAlg_Figure_02_01_016b_img.jpg
Combine like terms. CNX_IntAlg_Figure_02_01_016c_img.jpg
To clear decimals, multiply by 100. CNX_IntAlg_Figure_02_01_016d_img.jpg
Distribute. CNX_IntAlg_Figure_02_01_016e_img.jpg
Subtract 15 from both sides. CNX_IntAlg_Figure_02_01_016f_img.jpg
Simplify. CNX_IntAlg_Figure_02_01_016g_img.jpg
Divide by 30. CNX_IntAlg_Figure_02_01_016h_img.jpg
Simplify. CNX_IntAlg_Figure_02_01_016i_img.jpg
Check it yourself by substituting x=9 into the original equation.

Exercise 2.2.13A

Solve: 0.25n+0.05(n+5)=2.95.

Answer

n=9

Exercise 2.2.13B

Solve: 0.10d+0.05(d5)=2.15.

Answer

d=16

Key Concepts

  • How to determine whether a number is a solution to an equation
    1. Substitute the number in for the variable in the equation.
    2. Simplify the expressions on both sides of the equation.
    3. Determine whether the resulting equation is true.

      If it is true, the number is a solution.

      If it is not true, the number is not a solution.

  • How to Solve Linear Equations Using a General Strategy
    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 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.

  • How to Solve Equations with Fraction or Decimal Coefficients
    1. Find the least common denominator (LCD) of all the fractions and decimals (in fraction form) in the equation.
    2. Multiply both sides of the equation by that LCD. This clears the fractions and decimals.
    3. Solve using the General Strategy for Solving Linear Equations.

Glossary

conditional equation
An equation that is true for one or more values of the variable and false for all other values of the variable is a conditional equation.
contradiction
An equation that is false for all values of the variable is called a contradiction. A contradiction has no solution.
identity
An equation that is true for any value of the variable is called an Identity. The solution of an identity is all real numbers.
linear equation
A linear equation is an equation in one variable that can be written, where a and b are real numbers and a0, as ax+b=0.
solution of an equation
A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

This page titled 2.2: Use a General Strategy to Solve Linear Equations is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

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