[latex]\begin{array}{}\\ \hfill x+2=7\hfill \\ \hfill 5+2\stackrel{?}{=}7\hfill \\ \\ \hfill 7=7\quad\checkmark \hfill \end{array}[/latex]
Since [latex]5+2=7[/latex] is a true statement, we know that [latex]5[/latex] is indeed a solution to the equation.
The symbol [latex]\stackrel{?}{=}[/latex] asks whether the left side of the equation is equal to the right side. Once we know, we can change to an equal sign [latex]=[/latex] or not-equal sign [latex]\not=[/latex].
example
Determine whether [latex]x=5[/latex] is a solution of [latex]6x - 17=16[/latex].
Solution
|
[latex]6x--17=16[/latex] |
| Substitute [latex]\color{red}{5}[/latex] for x. |
[latex]6\cdot\color{red}{5}--17=16[/latex] |
| Multiply. |
[latex]30--17=16[/latex] |
| Subtract. |
[latex]13\not=16[/latex] |
So [latex]x=5[/latex] is not a solution to the equation [latex]6x - 17=16[/latex].
example
Determine whether [latex]y=2[/latex] is a solution of [latex]6y - 4=5y - 2[/latex].
Answer:
Solution
Here, the variable appears on both sides of the equation. We must substitute [latex]2[/latex] for each [latex]y[/latex].
|
[latex]6y--4=5y--2[/latex] |
| Substitute [latex]\color{red}{2}[/latex] for y. |
[latex]6(\color{red}{2})--4=5(\color{red}{2})--2[/latex] |
| Multiply. |
[latex]12--4=10--2[/latex] |
| Subtract. |
[latex]8=8\quad\checkmark[/latex] |
Since [latex]y=2[/latex] results in a true equation, we know that [latex]2[/latex] is a solution to the equation [latex]6y - 4=5y - 2[/latex].
example
Determine whether each of the following is a solution of [latex]2x - 5=-13\text{:}[/latex]
1. [latex]x=4[/latex]
2. [latex]x=-4[/latex]
3. [latex]x=-9[/latex]
Solution
| 1. Substitute [latex]4[/latex] for x in the equation to determine if it is true. |
|
|
[latex]2x--5=--13[/latex] |
| Substitute [latex]\color{red}{4}[/latex] for x. |
[latex]2(\color{red}{4})--5=--13[/latex] |
| Multiply. |
[latex]8--5=--13[/latex] |
| Subtract. |
[latex]3\not=--13[/latex] |
Since [latex]x=4[/latex] does not result in a true equation, [latex]4[/latex] is not a solution to the equation.
| 2. Substitute [latex]−4[/latex] for x in the equation to determine if it is true. |
|
|
[latex]2x--5=--13[/latex] |
| Substitute [latex]\color{red}{--4}[/latex] for x. |
[latex]2(\color{red}{-4})--5=--13[/latex] |
| Multiply. |
[latex]--8--5=--13[/latex] |
| Subtract. |
[latex]--13=--13\quad\checkmark[/latex] |
Since [latex]x=-4[/latex] results in a true equation, [latex]-4[/latex] is a solution to the equation.
| 3. Substitute [latex]−9[/latex] for x in the equation to determine if it is true. |
|
|
[latex]2x--5=--13[/latex] |
| Substitute [latex]−9[/latex] for x. |
[latex]2(\color{red}{--9})--5=--13[/latex] |
| Multiply. |
[latex]--18--5=--13[/latex] |
| Subtract. |
[latex]--23\not=--13[/latex] |
Since [latex]x=-9[/latex] does not result in a true equation, [latex]-9[/latex] is not a solution to the equation.
