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}}=\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].
Answer:
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].
Answer:
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] |
Example
Solve: [latex]8 - 2\left(3y+5\right)=0[/latex].
Answer:
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}})+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].
Answer:
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] |
Example
Solve: [latex]\frac{1}{2}\left(6x - 2\right)=5-x[/latex].
Answer:
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] |
|
[latex]\frac{7}{2}=\frac{7}{2}\quad\checkmark[/latex] |
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: [latex]0.45\left(a+0.8\right)=0.3\left(a+2.2\right)[/latex].
Answer:
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] |