Learning Outcomes
- Determine whether a number is a solution to an equation
- Check your solution to a linear equation to verify its accuracy
- Solve equations using the Subtraction and Addition Properties of Equality
- Solve equations that need to be simplified
We began our work solving equations in previous chapters, where we said that solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle. When you solve an equation, you find the value of the variable that makes the equation true.
The simplest type of algebraic equation is a linear equation that has just one variable. We will be solving this type of equation in this section. Specifically, you’ll learn how to use the Subtraction and Addition Properties of Equality. But first, we will review how to determine whether a number is a solution of an equation.
Determine Whether a Number is a Solution of an Equation
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=\Large\frac{3}{4}[/latex] is a solution for [latex]4y+3=8y[/latex].
Solution:
| [latex]4y+3=8y[/latex] | |
| Substitute [latex]\color{red}{\Large\frac{3}{4}}[/latex] for [latex]y[/latex] | [latex]4(\color{red}{\Large\frac{3}{4}}\normalsize)+3\stackrel{\text{?}}{=}8(\color{red}{\Large\frac{3}{4}})[/latex] |
| Multiply. | [latex]3+3\stackrel{\text{?}}{=}6[/latex] |
| Add. | [latex]6=6\quad\checkmark[/latex] |
Since [latex]y=\Large\frac{3}{4}[/latex] results in a true equation, [latex]\Large\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.
TRY IT
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Subtraction and Addition Properties of Equality
We introduced the Subtraction and Addition Properties of Equality earlier by modeling equations with envelopes and counters. When you add or subtract the same quantity from both sides of an equation, you still have equality. The image below models this with the equation [latex]x+3=8[/latex].
The goal is to isolate the variable on one side of the equation. So we “took away” [latex]3[/latex] from both sides of the equation and found the solution [latex]x=5[/latex].
Some people picture a balance scale, as in the image below, when they solve equations.
The quantities on both sides of the equal sign in an equation are equal, or balanced. Just as with the balance scale, whatever you do to one side of the equation you must also do to the other, to keep it balanced. The Addition and Subtraction Properties of Equality state that you can add or subtract the same quantity to both sides of an equation, and still maintain an equivalent equation. Sometimes people refer to this as keeping the equation “balanced.” If you think of an equation as being like a balance scale, the quantities on each side of the equation are equal, or balanced.
Let’s look at a simple numeric equation, [latex]3+7=10[/latex], to explore the idea of an equation as being balanced.
The expressions on each side of the equal sign are equal, so you can add the same value to each side and maintain the equality. Let’s see what happens when [latex]5[/latex] is added to each side.
[latex]3+7+5=10+5[/latex]
[latex]15=15[/latex]
Since each expression is equal to [latex]15[/latex], you can see that adding [latex]5[/latex] to each side of the original equation resulted in a true equation. The equation is still “balanced.”
On the other hand, let’s look at what would happen if you added [latex]5[/latex] to only one side of the equation.
[latex]\begin{array}{r}3+7=10\\3+7+5=10\\15\neq 10\end{array}[/latex]
Adding [latex]5[/latex] to only one side of the equation resulted in an equation that is false. The equation is no longer “balanced,” and it is no longer a true equation!
Subtraction Property of Equality
For all real numbers [latex]a,b[/latex], and [latex]c[/latex], if [latex]a=b[/latex], then [latex]a-c=b-c[/latex].
If two expressions are equal to each other, and you subtract the same value to both sides of the equation, the equation will remain equal.
Addition Property of Equality
For all real numbers [latex]a,b[/latex], and [latex]c[/latex], if [latex]a=b[/latex], then [latex]a+c=b+c[/latex].
If two expressions are equal to each other, and you add the same value to both sides of the equation, the equation will remain equal.
In order to solve an equation, you isolate the variable. Isolating the variable means rewriting an equivalent equation in which the variable is on one side of the equation and everything else is on the other side of the equation. When you follow the steps to solve an equation, you try to isolate the variable. The variable is a quantity we don’t know yet. You have a solution when you get the equation [latex]x[/latex] = some value.
When the equation involves addition or subtraction, use the inverse operation to “undo” the operation in order to isolate the variable. For addition and subtraction, your goal is to change any value being added or subtracted to [latex]0[/latex], the additive identity.
In the following examples, we use Subtraction and Addition Properties of Equality to solve equations. We need to isolate the variable on one side of the equation. You can check your solutions by substituting the value into the equation, to make sure you have a true statement.
EXAMPLE
Solve: [latex]x+11=-3[/latex]
Now you can try solving an equation that requires using the Subtraction Property of Equality.
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]
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.
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-\Large\frac{3}{8}\normalsize =\Large\frac{1}{2}[/latex]
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.
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]
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.
Think About It
Can you determine what you would do differently if you were asked to solve equations like these?
a) Solve [latex]{12.5}+{ t }= {-7.5}[/latex].
What makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would solve this equation with decimals.
b) Solve [latex]\frac{1}{4} + y = 3[/latex]. What makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would solve this equation with a fraction.
Solving Equations that Need to be Simplified
The examples above are sometimes called one-step equations because they require only one step to solve. In these examples, you either added or subtracted a constant from both sides of the equation to isolate the variable and solve the equation.
In the examples up to this point, we have been able to isolate the variable with just one operation. Many of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation before using the Subtraction or Addition Properties of Equality. You should always simplify as much as possible before trying to isolate the variable.
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]3x-7-2x-4=1[/latex][latex]3(\color{red}{12})-7-2(\color{red}{12})-4=1[/latex][latex]36-7-24-4=1[/latex][latex]29-24-4=1[/latex][latex]5-4=1[/latex][latex]1=1\quad\checkmark[/latex]The solution checks. |
Now you can try solving a couple of equations where you should simplify first.
