Learning Outcomes
- Solve a linear equation that requires multiple steps and a combination of the properties of equality
- Solve equations with fraction coefficients
- Solve equations with decimal coefficients
Key WORDS
- General Strategy: a plan that can be followed that works in all cases
It’s time now to lay out an overall strategy that can be used to solve any linear equation in one variable. 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 in one variable
- 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 equal to [latex]1[/latex]. Use the Multiplication or Division Property of Equality.
- State the solution to the equation. If there is a contradiction, there is no solution. If there is an identity, the solution is the set of all real numbers.
- 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]\frac{3x}{\color{red}{3}}\normalsize =\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] |
Answer: [latex]x=4[/latex]
Example
Solve: [latex]-\left(x+5\right)=7[/latex]
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] |
Answer: [latex]x=-12[/latex]
Example
Solve: [latex]4\left(x - 2\right)+5=-3[/latex]
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]\frac{4x}{\color{red}{4}} =\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] |
Answer: [latex]x=0[/latex]
Example
Solve: [latex]8 - 2\left(3y+5\right)=0[/latex]
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]\\frac{-6y}{\color{red}{-6}} =\frac{2}{\color{red}{-6}}[/latex] |
| Simplify. | [latex]y=-\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}{-\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] |
Answer: [latex]y=-\frac{1}{3}[/latex]
example
Solve: [latex]3\left(x - 2\right)-5=4\left(2x+1\right)+5[/latex]
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]\frac{-20}{\color{red}{5}} =\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] |
Answer: [latex]x=-4[/latex]
try it
https://ohm.lumenlearning.com/multiembedq.php?id=142586&theme=oea&iframe_resize_id=mom5
https://ohm.lumenlearning.com/multiembedq.php?id=142587&theme=oea&iframe_resize_id=mom6
Example
Solve: [latex]\frac{1}{2}\left(6x - 2\right)=5-x[/latex]
Solution:
| [latex]\frac{1}{2}(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]\frac{4x}{\color{red}{4}} =\frac{6}{\color{red}{4}}[/latex] |
| Simplify. | [latex]x=\frac{3}{2}[/latex] |
| Check: Let [latex]x=\frac{3}{2}[/latex] . | [latex]\frac{1}{2} (6(\frac{\color{red}{3}}{\color{red}{2}} )-2)\overset{?}{=}5-(\frac{\color{red}3}{\color{red}2})[/latex] |
| [latex]\frac{1}{2}(9-2)\overset{?}{=}\frac{10}{2} -\frac{3}{2}[/latex] | |
| [latex]\frac{1}{2}(7)\overset{?}{=\frac{7}{2}[/latex] | |
| [latexe\frac{7}{2} =\frac{7}{2}\quad\checkmark[/latex] |
Answer: [latex]x=\frac{3}{2}[/latex]
try it
https://ohm.lumenlearning.com/multiembedq.php?id=142589&theme=oea&iframe_resize_id=mom7
https://ohm.lumenlearning.com/multiembedq.php?id=142581&theme=oea&iframe_resize_id=mom8
https://ohm.lumenlearning.com/multiembedq.php?id=142582&theme=oea&iframe_resize_id=mom9
Watch the following video to see another example of how to solve an equation that requires distributing a fraction.
In many applications, we will have to solve equations with decimals. The same general strategy will work for these equations. We can choose to work with the decimals, or to clear the decimals. To clear decimals, we multiply both sides of the equation by an appropriate power of 10. The power is determined by the decimal with the most decimal places after the point.
example
Solve: [latex]0.45\left(a+0.8\right)=0.3\left(a+2.2\right)[/latex]
Solution:
Since the longest decimal has 2 decimal places after the point, we multiply by 100.
| [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]a = 2[/latex] |
| Check: Let [latex]a=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] |
Answer: [latex]a = 2[/latex]
try it
The following video provides another example of how to solve an equation that requires distributing a decimal.
Example
Solve:
1. [latex]5(4-x)=-3(x+6)-2x[/latex]
2. [latex]3-(5x-7)=2(5-7x)+9x[/latex]
Solution:
1.
[latex]\begin{equation}\begin{aligned}5(4-x)&=-3(x+6)-2x \;\;\;\;\;\;\;\;\;\;\text{Distribute} \\20-5x&=-3x-18-2x\;\;\;\;\;\;\;\;\;\;\text{Combine like terms}\\20-5x&=-5x-18\;\;\;\;\;\;\;\;\;\;\text{Add 5x to both sides}\\20-5x+5x&=-18\;\;\;\;\;\;\;\;\;\;\text{Combine like terms}\\ 20&=-18\end{aligned}\end{equation}[/latex]
Contradiction. No solution.
2.
[latex]\begin{equation}\begin{aligned}3-(5x-7)&=2(5-7x)+9x\;\;\;\;\;\;\;\;\;\;\text{Distribute}\\ 3-5x+7&=10-14x+9x\;\;\;\;\;\;\;\;\;\;\text{Combine like terms}\\ 10-5x&=10-5x\end{aligned}\end{equation}[/latex]
Identity. Solution is all real numbers.
Try It
Solve:
1. [latex]3(x+7)=2(x+6)+x[/latex]
2. [latex]9x-(3x-7)=2(3x-4)+15[/latex]
