# 2.3 Solving equations using the subtraction and addition properties  (Page 3/6)

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Think about twin brothers Andy and Bobby. They are $17$ years old. How old was Andy $3$ years ago? He was $3$ years less than $17,$ so his age was $17-3,$ or $14.$ What about Bobby’s age $3$ years ago? Of course, he was $14$ also. Their ages are equal now, and subtracting the same quantity from both of them resulted in equal ages $3$ years ago.

$\begin{array}{c}a=b\\ a-3=b-3\end{array}$

## Solve an equation using the subtraction property of equality.

1. Use the Subtraction Property of Equality to isolate the variable.
2. Simplify the expressions on both sides of the equation.
3. Check the solution.

Solve: $x+8=17.$

## Solution

We will use the Subtraction Property of Equality to isolate $x.$

 Subtract 8 from both sides. Simplify.

Since $x=9$ makes $x+8=17$ a true statement, we know $9$ is the solution to the equation.

Solve:

$x+6=19$

x = 13

Solve:

$x+9=14$

x = 5

Solve: $100=y+74.$

## Solution

To solve an equation, we must always isolate the variable—it doesn’t matter which side it is on. To isolate $y,$ we will subtract $74$ from both sides.

 Subtract 74 from both sides. Simplify. Substitute $26$ for $y$ to check.

Since $y=26$ makes $100=y+74$ a true statement, we have found the solution to this equation.

Solve:

$95=y+67$

y = 28

Solve:

$91=y+45$

y = 46

## Solve equations using the addition property of equality

In all the equations we have solved so far, a number was added to the variable on one side of the equation. We used subtraction to “undo” the addition in order to isolate the variable.

But suppose we have an equation with a number subtracted from the variable, such as $x-5=8.$ We want to isolate the variable, so to “undo” the subtraction we will add the number to both sides.

We use the Addition Property of Equality, which says we can add the same number to both sides of the equation without changing the equality. Notice how it mirrors the Subtraction Property of Equality.

## Addition property of equality

For any numbers $a,b,$ and $c,$ if

$a=b$

then

$a+c=b+c$

Remember the $17\text{-year-old}$ twins, Andy and Bobby? In ten years, Andy’s age will still equal Bobby’s age. They will both be $27.$

$\begin{array}{c}a=b\\ a+10=b+10\end{array}$

We can add the same number to both sides and still keep the equality.

## Solve an equation using the addition property of equality.

1. Use the Addition Property of Equality to isolate the variable.
2. Simplify the expressions on both sides of the equation.
3. Check the solution.

Solve: $x-5=8.$

## Solution

We will use the Addition Property of Equality to isolate the variable.

 Add 5 to both sides. Simplify.

Solve:

$x-9=13$

x = 22

Solve:

$y-1=3$

y = 4

Solve: $27=a-16.$

## Solution

We will add $16$ to each side to isolate the variable.

 Add 16 to each side. Simplify.

The solution to $27=a-16$ is $a=43.$

Solve:

$19=a-18$

a = 37

Solve:

$27=n-14$

n = 41

## Translate word phrases to algebraic equations

Remember, an equation has an equal sign between two algebraic expressions. So if we have a sentence that tells us that two phrases are equal, we can translate it into an equation. We look for clue words that mean equals . Some words that translate to the equal sign are:

• is equal to
• is the same as
• is
• gives
• was
• will be

It may be helpful to put a box around the equals word(s) in the sentence to help you focus separately on each phrase. Then translate each phrase into an expression, and write them on each side of the equal sign.

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