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
- Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.
- If there are fractions or decimals in the equation, multiply by the least common denominator to clear them.
- Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality.
- Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality.
- Make the coefficient of the variable term to equal to [latex]1[/latex]. Use the Multiplication or Division Property of Equality. State the solution to the equation.
- Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.
Example
Solve: [latex]3\left(x+2\right)=18[/latex]
Solution:
|
[latex]3(x+2)=18[/latex] |
| Simplify each side of the equation as much as possible.
Use the Distributive Property. |
[latex]3x+6=18[/latex] |
| Collect all variable terms on one side of the equation—all [latex]x[/latex] s are already on the left side. |
|
| Collect constant terms on the other side of the equation.
Subtract [latex]6[/latex] from each side. |
[latex]3x+6\color{red}{-6}=18\color{red}{-6}[/latex] |
| Simplify. |
[latex]3x=12[/latex] |
| Make the coefficient of the variable term equal to [latex]1[/latex]. Divide each side by [latex]3[/latex]. |
[latex]\Large\frac{3x}{\color{red}{3}}\normalsize =\Large\frac{12}{\color{red}{3}}[/latex] |
| Simplify. |
[latex]x=4[/latex] |
| Check: |
[latex]3(x+2)=18[/latex] |
| Let [latex]x=4[/latex]. |
[latex]3(\color{red}{4}+2)\stackrel{\text{?}}{=}18[/latex] |
|
[latex]3(6)\stackrel{\text{?}}{=}18[/latex] |
|
[latex]18=18\quad\checkmark[/latex] |
Example
Solve: [latex]-\left(x+5\right)=7[/latex]
Show Solution
Solution:
|
[latex]-(x+5)=7[/latex] |
| Simplify each side of the equation as much as possible by distributing.
The only [latex]x[/latex] term is on the left side, so all variable terms are on the left side of the equation. |
[latex]-x-5=7[/latex] |
| Add [latex]5[/latex] to both sides to get all constant terms on the right side of the equation. |
[latex]-x-5\color{red}{+5}=7\color{red}{+5}[/latex] |
| Simplify. |
[latex]-x=12[/latex] |
| Make the coefficient of the variable term equal to [latex]1[/latex] by multiplying both sides by [latex]-1[/latex]. |
[latex]\color{red}{-1}(-x)=\color{red}{-1}(12)[/latex] |
| Simplify. |
[latex]x=-12[/latex] |
| Check: |
[latex]-(x+5)=7[/latex] |
| Let [latex]x=-12[/latex]. |
[latex]-(\color{red}{-12}+5)\stackrel{\text{?}}{=}7[/latex] |
|
[latex]-(-7)\stackrel{\text{?}}{=}7[/latex] |
|
[latex]7=7\quad\checkmark[/latex] |
Example
Solve: [latex]4\left(x - 2\right)+5=-3[/latex]
Show Solution
Solution:
|
[latex]4(x-2)+5=-3[/latex] |
| Simplify each side of the equation as much as possible.
Distribute. |
[latex]4x-8+5=-3[/latex] |
| Combine like terms |
[latex]4x-3=-3[/latex] |
| The only [latex]x[/latex] is on the left side, so all variable terms are on one side of the equation. |
|
| Add [latex]3[/latex] to both sides to get all constant terms on the other side of the equation. |
[latex]4x-3\color{red}{+3}=-3\color{red}{+3}[/latex] |
| Simplify. |
[latex]4x=0[/latex] |
| Make the coefficient of the variable term equal to [latex]1[/latex] by dividing both sides by [latex]4[/latex]. |
[latex]\Large\frac{4x}{\color{red}{4}}\normalsize =\Large\frac{0}{\color{red}{4}}[/latex] |
| Simplify. |
[latex]x=0[/latex] |
| Check: |
[latex]4(x-2)+5=-3[/latex] |
| Let [latex]x=0[/latex]. |
[latex]4(\color{red}{0-2})+5\stackrel{\text{?}}{=}-3[/latex] |
|
[latex]4(-2)+5\stackrel{\text{?}}{=}-3[/latex] |
|
[latex]-8+5\stackrel{\text{?}}{=}-3[/latex] |
|
[latex]-3=-3\quad\checkmark[/latex] |
Example
Solve: [latex]8 - 2\left(3y+5\right)=0[/latex]
Show Solution
Solution:
Be careful when distributing the negative.
|
[latex]8-2(3y+5)=0[/latex] |
| Simplify—use the Distributive Property. |
[latex]8-6y-10=0[/latex] |
| Combine like terms. |
[latex]-6y-2=0[/latex] |
| Add [latex]2[/latex] to both sides to collect constants on the right. |
[latex]-6y-2\color{red}{+2}=0\color{red}{+2}[/latex] |
| Simplify. |
[latex]-6y=2[/latex] |
| Divide both sides by [latex]-6[/latex]. |
[latex]\Large\frac{-6y}{\color{red}{-6}}\normalsize =\Large\frac{2}{\color{red}{-6}}[/latex] |
| Simplify. |
[latex]y=-\Large\frac{1}{3}[/latex] |
| Check: |
[latex]8-2(3y+5)=0[/latex] |
| Let [latex]y=-\frac{1}{3}[/latex] |
[latex]8-2[3(\color{red}{-\Large\frac{1}{3}}\normalsize )+5]\stackrel{\text{?}}{=}0[/latex] |
|
[latex]8-2(-1+5)\stackrel{\text{?}}{=}0[/latex] |
|
[latex]8-2(4)\stackrel{\text{?}}{=}0[/latex] |
|
[latex]8-8\stackrel{\text{?}}{=}0[/latex] |
|
[latex]0=0\quad\checkmark[/latex] |
example
Solve: [latex]3\left(x - 2\right)-5=4\left(2x+1\right)+5[/latex]
Show Solution
Solution:
|
[latex]3(x-2)-5=4(2x+1)+5[/latex] |
| Distribute. |
[latex]3x-6-5=8x+4+5[/latex] |
| Combine like terms. |
[latex]3x-11=8x+9[/latex] |
| Subtract [latex]3x[/latex] to get all the variables on the right since [latex]8>3[/latex] . |
[latex]3x\color{red}{-3x}-11=8x\color{red}{-3x}+9[/latex] |
| Simplify. |
[latex]-11=5x+9[/latex] |
| Subtract [latex]9[/latex] to get the constants on the left. |
[latex]-11\color{red}{-9}=5x+9\color{red}{-9}[/latex] |
| Simplify. |
[latex]-20=5x[/latex] |
| Divide by [latex]5[/latex]. |
[latex]\Large\frac{-20}{\color{red}{5}}\normalsize =\Large\frac{5x}{\color{red}{5}}[/latex] |
| Simplify. |
[latex]-4=x[/latex] |
| Check: Substitute: [latex]-4=x[/latex] . |
[latex]3(\color{red}{-4}-2)-5\overset{?}{=}4(2(\color{red}{-4})+1)+5[/latex] |
|
[latex]3(-6)-5\overset{?}{=}4(-8+1)+5[/latex] |
|
[latex]-18-5\overset{?}{=}4(-7)+5[/latex] |
|
[latex]-23\overset{?}{=}-28+5[/latex] |
|
[latex]-23\overset{?}{=}-23\quad\checkmark[/latex] |
Example
Solve: [latex]\Large\frac{1}{2}\normalsize\left(6x - 2\right)=5-x[/latex]
Show Solution
Solution:
|
[latex]\Large\frac{1}{2}\normalsize(6x-2)=5-x[/latex] |
| Distribute. |
[latex]3x-1=5-x[/latex] |
| Add [latex]x[/latex] to get all the variables on the left. |
[latex]3x-1\color{red}{+x}=5-x\color{red}{+x}[/latex] |
| Simplify. |
[latex]4x-1=5[/latex] |
| Add [latex]1[/latex] to get constants on the right. |
[latex]4x-1\color{red}{+1}=5\color{red}{+1}[/latex] |
| Simplify. |
[latex]4x=6[/latex] |
| Divide by [latex]4[/latex]. |
[latex]\Large\frac{4x}{\color{red}{4}}\normalsize =\Large\frac{6}{\color{red}{4}}[/latex] |
| Simplify. |
[latex]x=\Large\frac{3}{2}[/latex] |
| Check: Let [latex]x=\Large\frac{3}{2}[/latex] . |
[latex]\Large\frac{1}{2}\normalsize (6(\Large\frac{\color{red}{3}}{\color{red}{2}}\normalsize )-2)\overset{?}{=}5-(\Large\frac{\color{red}3}{\color{red}2})[/latex] |
|
[latex]\Large\frac{1}{2}\normalsize(9-2)\overset{?}{=}\Large\frac{10}{2}\normalsize -\Large\frac{3}{2}[/latex] |
|
[latex]\Large\frac{1}{2}\normalsize(7)\overset{?}{=}\Large\frac{7}{2}[/latex] |
|
[latex]\Large\frac{7}{2}\normalsize =\Large\frac{7}{2}\normalsize\quad\checkmark[/latex] |
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: [latex]0.45\left(a+0.8\right)=0.3\left(a+2.2\right)[/latex]
Show Solution
Solution:
|
[latex]0.45\left(a+0.8\right)=0.3\left(a+2.2\right)[/latex] |
| Distribute. |
[latex]0.45a+0.36=0.3a+0.66[/latex] |
| Multiply by the least common denominator, 100 |
[latex]45a+36=30a+66[/latex] |
| Subtract [latex]30a[/latex] to get all the [latex]x[/latex] s to the left. |
[latex]45a\color{red}{-30a}+36=30a+66\color{red}{-30a}[/latex] |
| Simplify. |
[latex]15a+36=66[/latex] |
| Subtract [latex]36[/latex] to get the constants to the right. |
[latex]15a+36\color{red}{-36}=66\color{red}{-36}[/latex] |
| Simplify. |
[latex]15a=30[/latex] |
| Divide. |
[latex]\large{\frac{15a}{15}=\frac{30}{15}}[/latex] |
| Simplify. |
[latex]x = 2[/latex] |
| Check: Let [latex]x=2[/latex] |
[latex]0.45\left(2+0.8\right)=0.3\left(2+2.2\right)[/latex] |
|
[latex]1.26=1.26\quad\checkmark[/latex] |
The following video provides another example of how to solve an equation that requires distributing a decimal.
