Examples
Solve: [latex]4x+6=-14[/latex]
Solution:
In this equation, the variable is only on the left side. It makes sense to call the left side the variable side. Therefore, the right side will be the constant side.
| Since the left side is the variable side, the 6 is out of place. We must "undo" adding [latex]6[/latex] by subtracting [latex]6[/latex], and to keep the equality we must subtract [latex]6[/latex] from both sides. Use the Subtraction Property of Equality. | [latex]4x+6\color{red}{-6}=-14\color{red}{-6}[/latex] |
| Simplify. | [latex]4x=-20[/latex] |
| Now all the [latex]x[/latex] s are on the left and the constant on the right. | |
| Use the Division Property of Equality. | [latex]\Large\frac{4x}{\color{red}{4}}\normalsize =\Large\frac{-20}{\color{red}{4}}[/latex] |
| Simplify. | [latex]x=-5[/latex] |
| Check: | [latex]4x+6=-14[/latex] |
| Let [latex]x=-5[/latex] . | [latex]4(\color{red}{-5})+6=-14[/latex] |
| [latex]-20+6=-14[/latex] |
| [latex]-14=-14\quad\checkmark[/latex] |
Solve: [latex]2y - 7=15[/latex]
Answer:
Solution:
Notice that the variable is only on the left side of the equation, so this will be the variable side and the right side will be the constant side. Since the left side is the variable side, the [latex]7[/latex] is out of place. It is subtracted from the [latex]2y[/latex], so to "undo" subtraction, add [latex]7[/latex] to both sides.
| [latex]2y-7[/latex] is the side containing a variable.[latex]15[/latex] is the side containing only a constant.
|
| Add [latex]7[/latex] to both sides. | [latex]2y-7\color{red}{+7}=15\color{red}{+7}[/latex] |
| Simplify. | [latex]2y=22[/latex] |
| The variables are now on one side and the constants on the other. |
| Divide both sides by [latex]2[/latex]. | [latex]\Large\frac{2y}{\color{red}{2}}\normalsize =\Large\frac{22}{\color{red}{2}}[/latex] |
| Simplify. | [latex]y=11[/latex] |
| Check: | [latex]2y-7=15[/latex] |
| Let [latex]y=11[/latex] . | [latex]2\cdot\color{red}{11}-7\stackrel{\text{?}}{=}15[/latex] |
| [latex]22-7\stackrel{\text{?}}{=}15[/latex] |
| [latex]15=15\quad\checkmark[/latex] |
ExampleS
Solve: [latex]5x=4x+7[/latex]
Answer:
Solution:
| [latex]5x[/latex] is the side containing only a variable.[latex]4x+7[/latex] is the side containing a constant.
| |
| We don't want any variables on the right, so subtract the [latex]4x[/latex] . | [latex]5x\color{red}{-4x}=4x\color{red}{-4x}+7[/latex] |
| Simplify. | [latex]x=7[/latex] |
| We have all the variables on one side and the constants on the other. We have solved the equation. | |
| Check: | | [latex]5x=4x+7[/latex] |
| Substitute [latex]7[/latex] for [latex]x[/latex] . | | [latex]5(\color{red}{7})\stackrel{\text{?}}{=}4(\color{red}{7})+7[/latex] |
| | [latex]35\stackrel{\text{?}}{=}28+7[/latex] |
| | [latex]35=35\quad\checkmark[/latex] |
Solve: [latex]7x=-x+24[/latex].
Answer:
Solution:
The only constant, [latex]24[/latex], is on the right, so let the left side be the variable side.
| [latex]7x[/latex] is the side containing only a variable.[latex]-x+24[/latex] is the side containing a constant.
| |
| Remove the [latex]-x[/latex] from the right side by adding [latex]x[/latex] to both sides. | [latex]7x\color{red}{+x}=-x\color{red}{+x}+24[/latex] |
| Simplify. | [latex]8x=24[/latex] |
| All the variables are on the left and the constants are on the right. Divide both sides by [latex]8[/latex]. | [latex]\Large\frac{8x}{\color{red}{8}}\normalsize =\Large\frac{24}{\color{red}{8}}[/latex] |
| Simplify. | [latex]x=3[/latex] |
| Check: | [latex]7x=-x+24[/latex] |
| Substitute [latex]x=3[/latex]. | [latex]7(\color{red}{3})\stackrel{\text{?}}{=}-(\color{red}{3})+24[/latex] |
| [latex]21=21\quad\checkmark[/latex] |
example
Solve: [latex]5y - 8=7y[/latex]
Answer:
Solution:
The only constant, [latex]-8[/latex], is on the left side of the equation, and the variable, [latex]y[/latex], is on both sides. Let’s leave the constant on the left and collect the variables to the right.
| [latex]5y-8[/latex] is the side containing a constant.[latex]7y[/latex] is the side containing only a variable.
| |
| Subtract [latex]5y[/latex] from both sides. | [latex]5y\color{red}{-5y}-8=7y\color{red}{-5y}[/latex] |
| Simplify. | [latex]-8=2y[/latex] |
| We have the variables on the right and the constants on the left. Divide both sides by [latex]2[/latex]. | [latex]\Large\frac{-8}{\color{red}{2}}\normalsize =\Large\frac{2y}{\color{red}{2}}[/latex] |
| Simplify. | [latex]-4=y[/latex] |
| Rewrite with the variable on the left. | [latex]y=-4[/latex] |
| Check: | [latex]5y-8=7y[/latex] |
| Let [latex]y=-4[/latex]. | [latex]5(\color{red}{-4})-8\stackrel{\text{?}}{=}7(\color{red}{-4})[/latex] |
| [latex]-20-8\stackrel{\text{?}}{=}-28[/latex] |
| [latex]-28=-28\quad\checkmark[/latex] |
Examples
Solve: [latex]7x+5=6x+2[/latex]
Answer:
Solution:
Start by choosing which side will be the variable side and which side will be the constant side. The variable terms are [latex]7x[/latex] and [latex]6x[/latex]. Since [latex]7[/latex] is greater than [latex]6[/latex], make the left side the variable side and so the right side will be the constant side.
| [latex]7x+5=6x+2[/latex] |
| Collect the variable terms to the left side by subtracting [latex]6x[/latex] from both sides. | [latex]7x\color{red}{-6x}+5=6x\color{red}{-6x}+2[/latex] |
| Simplify. | [latex]x+5=2[/latex] |
| Now, collect the constants to the right side by subtracting [latex]5[/latex] from both sides. | [latex]x+5\color{red}{-5}=2\color{red}{-5}[/latex] |
| Simplify. | [latex]x=-3[/latex] |
| The solution is [latex]x=-3[/latex] . | |
| Check: | [latex]7x+5=6x+2[/latex] |
| Let [latex]x=-3[/latex]. | [latex]7(\color{red}{-3})+5\stackrel{\text{?}}{=}6(\color{red}{-3})+2[/latex] |
| [latex]-21+5\stackrel{\text{?}}{=}-18+2[/latex] |
| [latex]16=16\quad\checkmark[/latex] |
Solve: [latex]6n - 2=-3n+7[/latex]
Answer:
We have [latex]6n[/latex] on the left and [latex]-3n[/latex] on the right. Since [latex]6>-3[/latex], make the left side the "variable" side.
| [latex]6n-2=-3n+7[/latex] |
| We don't want variables on the right side—add [latex]3n[/latex] to both sides to leave only constants on the right. | [latex]6n\color{red}{+3n}-2=-3n\color{red}{+3n}+7[/latex] |
| Combine like terms. | [latex]9n-2=7[/latex] |
| We don't want any constants on the left side, so add [latex]2[/latex] to both sides. | [latex]9n-2\color{red}{+2}=7\color{red}{+2}[/latex] |
| Simplify. | [latex]9n=9[/latex] |
| The variable term is on the left and the constant term is on the right. To get the coefficient of [latex]n[/latex] to be one, divide both sides by [latex]9[/latex]. | [latex]\Large\frac{9n}{\color{red}{9}}\normalsize =\Large\frac{9}{\color{red}{9}}[/latex] |
| Simplify. | [latex]n=1[/latex] |
| Check: | [latex]6n-2=-3n+7[/latex] |
| Substitute [latex]1[/latex] for [latex]n[/latex]. | [latex]6(\color{red}{1})-2\stackrel{\text{?}}{=}-3(\color{red}{1})+7[/latex] |
| [latex]4=4\quad\checkmark[/latex] |
EXAMPLE
Solve: [latex]2a - 7=5a+8[/latex]
Answer:
Solution:
This equation has [latex]2a[/latex] on the left and [latex]5a[/latex] on the right. Since [latex]5>2[/latex], make the right side the variable side and the left side the constant side.
| [latex]2a-7=5a+8[/latex] |
| Subtract [latex]2a[/latex] from both sides to remove the variable term from the left. | [latex]2a\color{red}{-2a}-7=5a\color{red}{-2a}+8[/latex] |
| Combine like terms. | [latex]-7=3a+8[/latex] |
| Subtract [latex]8[/latex] from both sides to remove the constant from the right. | [latex]-7\color{red}{-8}=3a+8\color{red}{-8}[/latex] |
| Simplify. | [latex]-15=3a[/latex] |
| Divide both sides by [latex]3[/latex] to make [latex]1[/latex] the coefficient of [latex]a[/latex] . | [latex]\Large\frac{-15}{\color{red}{3}}\normalsize =\Large\frac{3a}{\color{red}{3}}[/latex] |
| Simplify. | [latex]-5=a[/latex] |
| Check: | [latex]2a-7=5a+8[/latex] |
| Let [latex]a=-5[/latex] | [latex]2(\color{red}{-5})-7\stackrel{\text{?}}{=}5(\color{red}{-5})+8[/latex] |
| [latex]-10-7\stackrel{\text{?}}{=}-25+8[/latex] |
| [latex]-17=-17\quad\checkmark[/latex] |
