Solution of an Equation
A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.
In the earlier sections, we listed the steps to determine if a value is a solution. We restate them here.
Determine whether a number is a solution to an equation.
- Substitute the number for the variable in the equation.
- Simplify the expressions on both sides of the equation.
- Determine whether the resulting equation is true.
- If it is true, the number is a solution.
- If it is not true, the number is not a solution.
In the following example, we will show how to determine whether a number is a solution to an equation that contains addition and subtraction. You can use this idea to check your work later when you are solving equations.
EXAMPLE
Determine whether [latex]y=\frac{3}{4}[/latex] is a solution for [latex]4y+3=8y[/latex].
Solution:
|
[latex]4y+3=8y[/latex] |
| Substitute [latex]\color{red}{\frac{3}{4}}[/latex] for [latex]y[/latex] |
[latex]4(\color{red}{\frac{3}{4}})+3\stackrel{\text{?}}{=}8(\color{red}{\frac{3}{4}})[/latex] |
| Multiply. |
[latex]3+3\stackrel{\text{?}}{=}6[/latex] |
| Add. |
[latex]6=6\quad\checkmark[/latex] |
Since [latex]y=\frac{3}{4}[/latex] results in a true equation, [latex]\frac{3}{4}[/latex] is a solution to the equation [latex]4y+3=8y[/latex].
Now it is your turn to determine whether a fraction is the solution to an equation.
We introduced the Subtraction and Addition Properties of Equality in Solving Equations Using the Subtraction and Addition Properties of Equality. In that section, we modeled how these properties work and then applied them to solving equations with whole numbers. We used these properties again each time we introduced a new system of numbers. Let’s review those properties here.
EXAMPLE
Solve: [latex]x+11=-3[/latex].
Answer:
Solution:
To isolate [latex]x[/latex], we undo the addition of [latex]11[/latex] by using the Subtraction Property of Equality.
|
[latex]x+11=-3[/latex] |
| Subtract 11 from each side to "undo" the addition. |
[latex]x+11\color{red}{-11}=-3\color{red}{-11}[/latex] |
| Simplify. |
[latex]x=-14[/latex] |
| Check: |
[latex]x+11=-3[/latex] |
|
| Substitute [latex]x=-14[/latex] . |
[latex]\color{red}{-14}+11\stackrel{\text{?}}{=}-3[/latex] |
|
|
[latex]-3=-3\quad\checkmark[/latex] |
|
Since [latex]x=-14[/latex] makes [latex]x+11=-3[/latex] a true statement, we know that it is a solution to the equation.
Now you can try solving an equation that requires using the addition property.
In the original equation in the previous example, [latex]11[/latex] was added to the [latex]x[/latex] , so we subtracted [latex]11[/latex] to "undo" the addition. In the next example, we will need to "undo" subtraction by using the Addition Property of Equality.
EXAMPLE
Solve: [latex]m - 4=-5[/latex].
Answer:
Solution:
|
[latex]m-4=-5[/latex] |
| Add 4 to each side to "undo" the subtraction. |
[latex]m-4\color{red}{+4}=-5\color{red}{+4}[/latex] |
| Simplify. |
[latex]m=-1[/latex] |
| Check: |
[latex]m-4=-5[/latex] |
|
| Substitute [latex]m=-1[/latex] . |
[latex]\color{red}{-1}+4\stackrel{\text{?}}{=}-5[/latex] |
|
|
[latex]-5=-5\quad\checkmark[/latex] |
|
|
|
The solution to [latex]m - 4=-5[/latex] is [latex]m=-1[/latex] . |
Now you can try using the addition property to solve an equation.
In the following video, we present more examples of solving equations using the addition and subtraction properties.
https://youtu.be/yqdlj0lv7Cc
You may encounter equations that contain fractions, therefore in the following examples we will demonstrate how to use the addition property of equality to solve an equation with fractions.
EXAMPLE
Solve: [latex]n-\frac{3}{8}=\frac{1}{2}[/latex].
Answer:
Solution:
|
[latex]n-\frac{3}{8}=\frac{1}{2}[/latex] |
| Use the Addition Property of Equality. |
[latex]n-\frac{3}{8}\color{red}{+\frac{3}{8}}=\frac{1}{2}\color{red}{+\frac{3}{8}}[/latex] |
| Find the LCD to add the fractions on the right. |
[latex]n-\frac{3}{8}+\frac{3}{8}=\frac{4}{8}+\frac{3}{8}[/latex] |
| Simplify. |
[latex]n=\frac{7}{8}[/latex] |
| Check: |
[latex]n-\frac{3}{8}=\frac{1}{2}[/latex] |
|
| Substitute [latex]n=\color{red}{\frac{7}{8}}[/latex] |
[latex]\color{red}{\frac{7}{8}}-\frac{3}{8}\stackrel{\text{?}}{=}\frac{1}{2}[/latex] |
|
| Subtract. |
[latex]\frac{4}{8}\stackrel{\text{?}}{=}\frac{1}{2}[/latex] |
|
| Simplify. |
[latex]\frac{1}{2}=\frac{1}{2}\quad\checkmark[/latex] |
|
| The solution checks. |
|
Now you can try solving an equation with fractions by using the addition property of equality.
Watch this video for more examples of solving equations that include fractions and require addition or subtraction.
https://youtu.be/KmOvCakGEgM
You may encounter equations with decimals, for example in financial or science applications. In the next examples we will demonstrate how to use the subtraction property of equality to solve equations with decimals.
eXAMPLE
Solve [latex]a - 3.7=4.3[/latex].
Answer:
Solution:
|
[latex]a-3.7=4.3[/latex] |
| Use the Addition Property of Equality. |
[latex]a-3.7\color{red}{+3.7}=4.3\color{red}{+3.7}[/latex] |
| Add. |
[latex]a=8[/latex] |
| Check: |
[latex]a-3.7=4.3[/latex] |
|
| Substitute [latex]a=8[/latex] . |
[latex]\color{red}{8}-3.7\stackrel{\text{?}}{=}4.3[/latex] |
|
| Simplify. |
[latex]4.3=4.3\quad\checkmark[/latex] |
|
| The solution checks. |
|
|
Now it is your turn to try solving an equation with decimals by using the addition property of equality.
In this video, we show more examples of how to solve equations with decimals that require addition and subtraction.
https://youtu.be/a6YYJN_bHKs
