example
Solve: [latex]4x=-28[/latex].
Solution:
To solve this equation, we use the Division Property of Equality to divide both sides by [latex]4[/latex].
| [latex]4x=-28[/latex] |
| Divide both sides by 4 to undo the multiplication. |
[latex]\frac{4x}{\color{red}4}=\frac{-28}{\color{red}4}[/latex] |
| Simplify. |
[latex]x =-7[/latex] |
| Check your answer. |
[latex]4x=-28[/latex] |
| Let [latex]x=-7[/latex]. Substitute [latex]-7[/latex] for x. |
[latex]4(\color{red}{-7})\stackrel{\text{?}}{=}-28[/latex] |
|
[latex]-28=-28[/latex] |
Since this is a true statement, [latex]x=-7[/latex] is a solution to [latex]4x=-28[/latex].
Now you can try to solve an equation that requires division and includes negative numbers.
In the previous example, to "undo" multiplication, we divided. How do you think we "undo" division? Next, we will show an example that requires us to use multiplication to undo division.
example
Solve: [latex]\frac{a}{-7}=-42[/latex].
Answer:
Solution:
Here [latex]a[/latex] is divided by [latex]-7[/latex]. We can multiply both sides by [latex]-7[/latex] to isolate [latex]a[/latex].
| [latex]\frac{a}{-7}=-42[/latex] |
| Multiply both sides by [latex]-7[/latex] . |
[latex]\color{red}{-7}(\frac{a}{-7})=\color{red}{-7}(-42)[/latex]
[latex]\frac{-7a}{-7}=294[/latex] |
| Simplify. |
[latex]a=294[/latex] |
| Check your answer. |
[latex]\frac{a}{-7}=-42[/latex] |
| Let [latex]a=294[/latex] . |
[latex]\frac{\color{red}{294}}{-7}\stackrel{\text{?}}{=}-42[/latex] |
|
[latex]-42=-42\quad\checkmark[/latex] |
Now see if you can solve a problem that requires multiplication to undo division. Recall the rules for multiplying two negative numbers - two negatives give a positive when they are multiplied.
As you begin to solve equations that require several steps you may find that you end up with an equation that looks like the one in the next example, with a negative variable. As a standard practice, it is good to ensure that variables are positive when you are solving equations. The next example will show you how.
example
Solve: [latex]-r=2[/latex].
Answer:
Solution:
Remember [latex]-r[/latex] is equivalent to [latex]-1r[/latex].
| [latex]-r=2[/latex] |
|
| Rewrite [latex]-r[/latex] as [latex]-1r[/latex] . |
[latex]-1r=2[/latex] |
| Divide both sides by [latex]-1[/latex] . |
[latex]\frac{-1r}{\color{red}{-1}}=\frac{2}{\color{red}{-1}}[/latex] |
| Simplify. |
[latex]r=-2[/latex] |
| Check. |
[latex]-r=2[/latex] |
| Substitute [latex]r=-2[/latex] |
[latex]-(\color{red}{-2})\stackrel{\text{?}}{=}2[/latex] |
| Simplify. |
[latex]2=2\quad\checkmark[/latex] |
Now you can try to solve an equation with a negative variable.
In our next example, we are given an equation that contains a variable multiplied by a fraction. We will use a reciprocal to isolate the variable.
example
Solve: [latex]\frac{2}{3}x=18[/latex].
Answer:
Solution:
Since the product of a number and its reciprocal is [latex]1[/latex], our strategy will be to isolate [latex]x[/latex] by multiplying by the reciprocal of [latex]\frac{2}{3}[/latex].
| [latex]\frac{2}{3}x=18[/latex] |
| Multiply by the reciprocal of [latex]\frac{2}{3}[/latex] . |
[latex]\frac{\color{red}{3}}{\color{red}{2}}\cdot\frac{2}{3}x[/latex] |
| Reciprocals multiply to one. |
[latex]1x=\frac{3}{2}\cdot\frac{18}{1}[/latex] |
| Multiply. |
[latex]x=27[/latex] |
| Check your answer. |
[latex]\frac{2}{3}x=18[/latex] |
| Let [latex]x=27[/latex]. |
[latex]\frac{2}{3}\cdot\color{red}{27}\stackrel{\text{?}}{=}18[/latex] |
|
[latex]18=18\quad\checkmark[/latex] |
Notice that we could have divided both sides of the equation [latex]\frac{2}{3}x=18[/latex] by [latex]\frac{2}{3}[/latex] to isolate [latex]x[/latex]. While this would work, multiplying by the reciprocal requires fewer steps.
The next video includes examples of using the division and multiplication properties to solve equations with the variable on the right side of the equal sign.
https://youtu.be/TB1rkPbF8rA