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+66=1863x+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=1818=18\quad\checkmark
 

Example

Solve:
(x+5)=7-\left(x+5\right)=7
.



 

Example

Solve:
4(x2)+5=34\left(x - 2\right)+5=-3
.



 

Example

Solve:
82(3y+5)=08 - 2\left(3y+5\right)=0
.



 

example

 

Solve:
3(x2)5=4(2x+1)+53\left(x - 2\right)-5=4\left(2x+1\right)+5
.



 

Example

Solve:
12(6x2)=5x\frac{1}{2}\left(6x - 2\right)=5-x
.



 

Watch the following video to see another example of how to solve an equation that requires distributing a fraction.



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.



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)
.



 

try it



The following video provides another example of how to solve an equation that requires distributing a decimal.



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