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Study Guides > Prealgebra

Solving Multi-Step Equations Using a General Strategy

Learning Outcomes

  • Identify the steps of a general problem solving strategy for solving linear equations
  • Use a general problem solving strategy to solve linear equations that require several steps
  Each of the first few sections of this chapter has dealt with solving one specific form of a linear equation. It’s time now to lay out an overall strategy that can be used to solve any linear equation. We call this the general strategy. Some equations won’t require all the steps to solve, but many will. Simplifying each side of the equation as much as possible first makes the rest of the steps easier.

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. If there are fractions or decimals in the equation, multiply by the least common denominator to clear them.
  3. Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality.
  4. Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality.
  5. Make the coefficient of the variable term to equal to 11. Use the Multiplication or Division Property of Equality. State the solution to the equation.
  6. Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.
 

Example

Solve: 3(x+2)=183\left(x+2\right)=18. Solution:
3(x+2)=183(x+2)=18
Simplify each side of the equation as much as possible. Use the Distributive Property. 3x+6=183x+6=18
Collect all variable terms on one side of the equation—all xx s are already on the left side.
Collect constant terms on the other side of the equation. Subtract 66 from each side. 3x+6−6=18−63x+6\color{red}{-6}=18\color{red}{-6}
Simplify. 3x=123x=12
Make the coefficient of the variable term equal to 11. Divide each side by 33. 3x3=123\frac{3x}{\color{red}{3}}=\frac{12}{\color{red}{3}}
Simplify. x=4x=4
Check:  3(x+2)=183(x+2)=18
Let x=4x=4. 3(4+2)=?183(\color{red}{4}+2)\stackrel{\text{?}}{=}18
3(6)=?183(6)\stackrel{\text{?}}{=}18
18=18✓18=18\quad\checkmark
   

Example

Solve: −(x+5)=7-\left(x+5\right)=7.

Answer:

Solution:
−(x+5)=7-(x+5)=7
Simplify each side of the equation as much as possible by distributing. The only xx term is on the left side, so all variable terms are on the left side of the equation. −x−5=7-x-5=7
Add 55 to both sides to get all constant terms on the right side of the equation. −x−5+5=7+5-x-5\color{red}{+5}=7\color{red}{+5}
Simplify. −x=12-x=12
Make the coefficient of the variable term equal to 11 by multiplying both sides by −1-1. −1(−x)=−1(12)\color{red}{-1}(-x)=\color{red}{-1}(12)
Simplify. x=−12x=-12
Check:  −(x+5)=7-(x+5)=7
Let x=−12x=-12.  −(−12+5)=?7-(\color{red}{-12}+5)\stackrel{\text{?}}{=}7
 −(−7)=?7-(-7)\stackrel{\text{?}}{=}7
7=7✓7=7\quad\checkmark

   

Example

Solve: 4(x−2)+5=−34\left(x - 2\right)+5=-3.

Answer:

Solution:
4(x−2)+5=−34(x-2)+5=-3
Simplify each side of the equation as much as possible. Distribute. 4x−8+5=−34x-8+5=-3
Combine like terms 4x−3=−34x-3=-3
The only xx is on the left side, so all variable terms are on one side of the equation.
Add 33 to both sides to get all constant terms on the other side of the equation. 4x−3+3=−3+34x-3\color{red}{+3}=-3\color{red}{+3}
Simplify. 4x=04x=0
Make the coefficient of the variable term equal to 11 by dividing both sides by 44. 4x4=04\frac{4x}{\color{red}{4}}=\frac{0}{\color{red}{4}}
Simplify. x=0x=0
Check:  4(x−2)+5=−34(x-2)+5=-3
Let x=0x=0. 4(0−2)+5=?−34(\color{red}{0-2})+5\stackrel{\text{?}}{=}-3
4(−2)+5=?−34(-2)+5\stackrel{\text{?}}{=}-3
 −8+5=?−3-8+5\stackrel{\text{?}}{=}-3
 −3=−3✓-3=-3\quad\checkmark

   

Example

Solve: 8−2(3y+5)=08 - 2\left(3y+5\right)=0.

Answer:

Solution: Be careful when distributing the negative.
8−2(3y+5)=08-2(3y+5)=0
Simplify—use the Distributive Property. 8−6y−10=08-6y-10=0
Combine like terms. −6y−2=0-6y-2=0
Add 22 to both sides to collect constants on the right. −6y−2+2=0+2-6y-2\color{red}{+2}=0\color{red}{+2}
Simplify. −6y=2-6y=2
Divide both sides by −6-6. −6y−6=2−6\frac{-6y}{\color{red}{-6}}=\frac{2}{\color{red}{-6}}
Simplify. y=−13y=-\frac{1}{3}
Check:  8−2(3y+5)=08-2(3y+5)=0
Let y=−13y=-\frac{1}{3}. 8−2[3(−13)+5]=?08-2[3(\color{red}{-\frac{1}{3}})+5]\stackrel{\text{?}}{=}0
8−2(−1+5)=?08-2(-1+5)\stackrel{\text{?}}{=}0
8−2(4)=?08-2(4)\stackrel{\text{?}}{=}0
 8−8=?08-8\stackrel{\text{?}}{=}0
0=0✓0=0\quad\checkmark

   

example

  Solve: 3(x−2)−5=4(2x+1)+53\left(x - 2\right)-5=4\left(2x+1\right)+5.

Answer:

Solution:
3(x−2)−5=4(2x+1)+53(x-2)-5=4(2x+1)+5
Distribute. 3x−6−5=8x+4+53x-6-5=8x+4+5
Combine like terms. 3x−11=8x+93x-11=8x+9
Subtract 3x3x to get all the variables on the right since 8>38>3 . 3x−3x−11=8x−3x+93x\color{red}{-3x}-11=8x\color{red}{-3x}+9
Simplify. −11=5x+9-11=5x+9
Subtract 99 to get the constants on the left. −11−9=5x+9−9-11\color{red}{-9}=5x+9\color{red}{-9}
Simplify. −20=5x-20=5x
Divide by 55. −205=5x5\frac{-20}{\color{red}{5}}=\frac{5x}{\color{red}{5}}
Simplify. −4=x-4=x
Check: Substitute: −4=x-4=x .  3(−4−2)−5=?4(2(−4)+1)+53(\color{red}{-4}-2)-5\overset{?}{=}4(2(\color{red}{-4})+1)+5
3(−6)−5=?4(−8+1)+53(-6)-5\overset{?}{=}4(-8+1)+5
−18−5=?4(−7)+5-18-5\overset{?}{=}4(-7)+5
−23=?−28+5-23\overset{?}{=}-28+5
−23=?−23✓-23\overset{?}{=}-23\quad\checkmark

   

Example

Solve: 12(6x−2)=5−x\frac{1}{2}\left(6x - 2\right)=5-x.

Answer:

Solution:
12(6x−2)=5−x\frac{1}{2}(6x-2)=5-x
Distribute. 3x−1=5−x3x-1=5-x
Add xx to get all the variables on the left. 3x−1+x=5−x+x3x-1\color{red}{+x}=5-x\color{red}{+x}
Simplify. 4x−1=54x-1=5
Add 11 to get constants on the right. 4x−1+1=5+14x-1\color{red}{+1}=5\color{red}{+1}
Simplify. 4x=64x=6
Divide by 44. 4x4=64\frac{4x}{\color{red}{4}}=\frac{6}{\color{red}{4}}
Simplify. x=32x=\frac{3}{2}
Check: Let x=32x=\frac{3}{2} .  12(6(32)−2)=?5−(32)\frac{1}{2}(6(\frac{\color{red}{3}}{\color{red}{2}})-2)\overset{?}{=}5-(\frac{\color{red}3}{\color{red}2})
12(9−2)=?102−32\frac{1}{2}(9-2)\overset{?}{=}\frac{10}{2}-\frac{3}{2}
12(7)=?72\frac{1}{2}(7)\overset{?}{=}\frac{7}{2}
72=72✓\frac{7}{2}=\frac{7}{2}\quad\checkmark

  Watch the following video to see another example of how to solve an equation that requires distributing a fraction. https://youtu.be/1dmEoG7DkN4 In the next video example we show an example of solving an equation that requires distributing a fraction.  In this case, you will need to clear fractions after you distribute. https://youtu.be/-P4KZECxo8Y In many applications, we will have to solve equations with decimals. The same general strategy will work for these equations.

example

Solve: 0.45(a+0.8)=0.3(a+2.2)0.45\left(a+0.8\right)=0.3\left(a+2.2\right).

Answer:

Solution:
0.45(a+0.8)=0.3(a+2.2)0.45\left(a+0.8\right)=0.3\left(a+2.2\right)
Distribute. 0.45a+0.36=0.3a+0.660.45a+0.36=0.3a+0.66
Multiply by the least common denominator, 100 45a+36=30a+6645a+36=30a+66
Subtract 30a30a to get all the xx s to the left. 45a−30a+36=30a+66−30a45a\color{red}{-30a}+36=30a+66\color{red}{-30a}
Simplify. 15a+36=6615a+36=66
Subtract 3636 to get the constants to the right. 15a+36−36=66−3615a+36\color{red}{-36}=66\color{red}{-36}
Simplify. 15a=3015a=30
Divide. 15a15=3015\large{\frac{15a}{15}=\frac{30}{15}}
Simplify. x=2x = 2
Check: Let x=2x=2  0.45(2+0.8)=0.3(2+2.2)0.45\left(2+0.8\right)=0.3\left(2+2.2\right)
 1.26=1.26✓1.26=1.26\quad\checkmark

 

try it

[ohm_question]140292[/ohm_question]
The following video provides another example of how to solve an equation that requires distributing a decimal. https://youtu.be/k0K8mat_EaI

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