Example
Solve:
[latex]3x - 7 - 2x - 4=1[/latex].
Solution:
The left side of the equation has an expression that we should simplify before trying to isolate the variable.
|
[latex]3x-7-2x-4=1[/latex] |
| Rearrange the terms, using the Commutative Property of Addition. |
[latex]3x-2x-7-4=1[/latex] |
| Combine like terms. |
[latex]x-11=1[/latex] |
| Add [latex]11[/latex] to both sides to isolate [latex]x[/latex] . |
[latex]x-11\color{red}{+11}=1\color{red}{+11}[/latex] |
| Simplify. |
[latex]x=12[/latex] |
| Check.
Substitute [latex]x=12[/latex] into the original equation.
[latex-display]3x-7-2x-4=1[/latex-display]
[latex-display]3(\color{red}{12})-7-2(\color{red}{12})-4=1[/latex-display]
[latex-display]36-7-24-4=1[/latex-display]
[latex-display]29-24-4=1[/latex-display]
[latex-display]5-4=1[/latex-display]
[latex-display]1=1\quad\checkmark[/latex-display]
The solution checks. |
|
Now you can try solving a couple equations where you should simplify first.
The last few examples involved simplifying using addition and subtraction. Let's look at an example where we need to distribute first in order to simplify the equation as much as possible.
example
Solve: [latex]3\left(n - 4\right)-2n=-3[/latex].
Answer:
Solution:
The left side of the equation has an expression that we should simplify.
|
[latex]3(n-4)-2n=-3[/latex] |
| Distribute on the left. |
[latex]3n-12-2n=-3[/latex] |
| Use the Commutative Property to rearrange terms. |
[latex]3n-2n-12=-3[/latex] |
| Combine like terms. |
[latex]n-12=-3[/latex] |
| Isolate n using the Addition Property of Equality. |
[latex]n-12\color{red}{+12}=-3\color{red}{+12}[/latex] |
| Simplify. |
[latex]n=9[/latex] |
| Check.
Substitute [latex]n=9[/latex] into the original equation.
[latex-display]3(n-4)-2n=-3[/latex-display]
[latex-display]3(\color{red}{9}-4)-2\cdot\color{red}{9}=-3[/latex-display]
[latex-display]3(5)-18=-3[/latex-display]
[latex-display]15-18=-3[/latex-display]
[latex-display]-3=-3\quad\checkmark[/latex-display]
The solution checks. |
|
Now you can try a few problems that involve distribution.
The next example has expressions on both sides that need to be simplified.
example
Solve: [latex]2\left(3k - 1\right)-5k=-2 - 7[/latex].
Answer:
Solution:
Both sides of the equation have expressions that we should simplify before we isolate the variable.
|
[latex]2(3k-1)-5k=-2-7[/latex] |
| Distribute on the left, subtract on the right. |
[latex]6k-2-5k=-9[/latex] |
| Use the Commutative Property of Addition. |
[latex]6k-5k-2=-9[/latex] |
| Combine like terms. |
[latex]k-2=-9[/latex] |
| Undo subtraction by using the Addition Property of Equality. |
[latex]k-2\color{red}{+2}=-9\color{red}{+2}[/latex] |
| Simplify. |
[latex]k=-7[/latex] |
| Check.
Let [latex]k=-7[/latex].
[latex-display]2(3k-1)-5k=-2-7[/latex-display]
[latex-display]2(3(\color{red}{-7}-1)-5(\color{red}{-7})=-2-7[/latex-display]
[latex-display]2(-21-1)-5(-7)=-9[/latex-display]
[latex-display]2(-22)+35=-9[/latex-display]
[latex-display]-44+35=-9[/latex-display]
[latex]-9=-9\quad\checkmark[/latex] |
The solution checks.
Now, you give it a try!
In the following video we present another example of how to solve an equation that requires simplifying before using the addition and subtraction properties.
https://youtu.be/shGKzDBA5kQ
