example
Determine whether each of the following is a solution of [latex]x - 0.7=1.5\text{:}[/latex]
ⓐ [latex]x=1[/latex] ⓑ [latex]x=-0.8[/latex] ⓒ [latex]x=2.2[/latex]
Solution
| ⓐ |
|
|
[latex]x-0.7=1.5[/latex] |
| Substitute [latex]\color{red}{1}[/latex] for x. |
[latex]\color{red}{1} - 0.7\stackrel{?}{=}1.5[/latex] |
| Subtract. |
[latex]0.3\color{red}{=}1.5[/latex] |
Since [latex]x=1[/latex] does not result in a true equation, [latex]1[/latex] is not a solution to the equation.
| ⓑ |
|
|
[latex]x-0.7=1.5[/latex] |
| Substitute [latex]\color{red}{0.8}[/latex] for x. |
[latex]\color{red}{0.8} - 0.7\stackrel{?}{=}1.5[/latex] |
| Subtract. |
[latex]-1.5\color{red}{=}1.5[/latex] |
Since [latex]x=-0.8[/latex] does not result in a true equation, [latex]-0.8[/latex] is not a solution to the equation.
| ⓒ |
|
|
[latex]x-0.7=1.5[/latex] |
| Substitute [latex]\color{red}{2.2}[/latex] for x. |
[latex]\color{red}{2.2} - 0.7\stackrel{?}{=}1.5[/latex] |
| Subtract. |
[latex]1.5=1.5[/latex] |
Since [latex]x=2.2[/latex] results in a true equation, [latex]2.2[/latex] is a solution to the equation.
example
Solve: [latex]y+2.3=-4.7[/latex].
Answer:
Solution
We will use the Subtraction Property of Equality to isolate the variable.
|
[latex]y+2.3=-4.7[/latex] |
| Subtract [latex]\color{red}{2.3}[/latex] from each side, to undo the addition. |
[latex]y+2.3\color{red}{- 2.3}=-4.7\color{red}{- 2.3}[/latex] |
| Simplify. |
[latex]y-7[/latex] |
| Check: |
[latex]y+2.3=-4.7[/latex] |
|
| Substitute [latex]y=\color{red}{-7}[/latex]. |
[latex]\color{red}{-7}+2.3\stackrel{?}{=}-4.7[/latex] |
|
| Simplify. |
[latex]-4.7=-4.7[/latex] |
|
Since [latex]y=-7[/latex] makes [latex]y+2.3=-4.7[/latex] a true statement, we know we have found a solution to this equation.
example
Solve: [latex]a - 4.75=-1.39[/latex].
Answer:
Solution
We will use the Addition Property of Equality.
|
[latex]a-4.75=-1.39[/latex] |
| Add [latex]4.75[/latex] to each side, to undo the subtraction. |
[latex]a-4.75+\color{red}{4.75}=-1.39+\color{red}{4.75}[/latex] |
| Simplify. |
[latex]a=3.36[/latex] |
| Check: |
[latex]a-4.75=-1.39[/latex] |
|
| Substitute [latex]a=\color{red}{3.36}[/latex]. |
[latex]\color{red}{3.36}-4.75\stackrel{?}{=}-1.39[/latex] |
|
|
[latex]-1.39=-1.39[/latex] |
|
Since the result is a true statement, [latex]a=3.36[/latex] is a solution to the equation.
example
Solve: [latex]-4.8=0.8n[/latex].
Answer:
Solution
We will use the Division Property of Equality.
Use the Properties of Equality to find a value for [latex]n[/latex].
|
[latex]-4.8=0.8n[/latex] |
| We must divide both sides by [latex]0.8[/latex] to isolate n. |
[latex]\frac{-4.8}{\color{red}{0.8}}=\frac{0.8n}{\color{red}{0.8}}[/latex] |
| Simplify. |
[latex]-6=n[/latex] |
| Check: |
[latex]-4.8=0.8n[/latex] |
|
| Substitute [latex]n=\color{red}{-6}[/latex]. |
[latex]-4.8\stackrel{?}{=}0.8(\color{red}{-6})[/latex] |
|
|
[latex]-4.8=-4.8[/latex] |
|
Since [latex]n=-6[/latex] makes [latex]-4.8=0.8n[/latex] a true statement, we know we have a solution.
example
Solve: [latex]\frac{p}{-1.8}=-6.5[/latex].
Answer:
Solution
We will use the Multiplication Property of Equality.
|
[latex]\frac{p}{-1.8}=-6.5[/latex] |
| Here, p is divided by [latex]−1.8[/latex]. We must multiply by [latex]−1.8[/latex] to isolate p |
[latex]\color{red}{-1.8}(\frac{p}{-1.8})=\color{red}{-1.8}(-6.5)[/latex] |
| Multiply. |
[latex]p=11.7[/latex] |
| Check: |
[latex]\frac{p}{-1.8}=-6.5[/latex] |
|
|
[latex]\frac{\color{red}{11.7}}{-1.8}\stackrel{?}{=}-6.5[/latex] |
|
| Substitute [latex]p=\color{red}{11.7}[/latex]. |
[latex]-6.5=-6.5[/latex] |
|
A solution to [latex]\frac{p}{-1.8}=-6.5[/latex] is [latex]p=11.7[/latex].
