# 7.1: Solve for the Unknown

**At Grade**Created by: CK-12

Can you use the order of operations? What's the value of \begin{align*}a\end{align*}? In this section, we will learn to use the order of operations.

\begin{align*}(3 + 5) \div 2 + 4 \times 6 = a\end{align*}

### Solve for the Unknown

The **order of operations** tells us the correct order of evaluating math expressions. We always do ** parenthesis first**. Then we do

**and finally**

*multiplication and division (from left to right)***.**

*addition and subtraction (from left to right)*In order to evaluate expressions using the order of operations, we can use the problem solving steps to help.

- First,
**describe**what you see in the problem. What operations are there? - Second, identify what
**your job**is. In these problems, your job will be to solve for the unknown. - Third, make a
**plan**. In these problems, your plan should be to use the order of operations. - Fourth,
**solve**. - Fifth,
**check**.

#### Solving for Unknown Values

1. Figure out the value of the variable. Follow the order of operations.

\begin{align*}8-(3-1)+6\times 2=e\end{align*}

We can use the problem solving steps to help.

**Describe:** The equation has parenthesis, multiplication, addition and subtraction.

**My Job:** Do the operations to figure out the value of \begin{align*}e\end{align*}.

**Plan:** Do the parenthesis first. Then do the multiplication. Finally, do the subtraction and addition from left to right.

**Solve:** First do the parenthesis: \begin{align*}8-(3-1)+6\times 2=8-2+6\times 2\end{align*}

- Next do the multiplication: \begin{align*}8-2+6\times 2=8-2+12\end{align*}
- Last do the subtraction and addition: \begin{align*}8-2+12=6+12=18\end{align*}
- \begin{align*}e=18\end{align*}

**Check:** \begin{align*}8-(3-1)+6\times 2=8-2+6\times 2=8-2+12=6+12=18\end{align*}

2. Figure out the value of the variable. Follow the order of operations.

\begin{align*}(4 + 6) \div 5 - 3 \div 3 = a\end{align*}

We can use the problem solving steps to help.

**Describe:** The equation has parenthesis, division, addition and subtraction.

**My Job:** Do the operations to figure out the value of \begin{align*}a\end{align*}.

**Plan:** Do the parenthesis first. Then do the division. Finally, do the subtraction and addition from left to right.

**Solve:** First do the parenthesis: \begin{align*}(4 + 6) \div 5 - 3 \div 3 =10\div 5-3 \div 3\end{align*}

- Next do the division: \begin{align*}10\div 5-3 \div 3=2-1\end{align*}
- Last do the subtraction: \begin{align*}2-1=1\end{align*}
- \begin{align*}a=1\end{align*}

**Check:** \begin{align*}(4 + 6) \div 5 - 3 \div 3 =10\div 5-3 \div 3=2-1=1\end{align*}

3. Figure out the value of the variable. Follow the order of operations.

\begin{align*}6 \times 7 \div 6 + 8 = c\end{align*}

We can use the problem solving steps to help.

**Describe:** The equation has multiplication, division and addition.

**My Job:** Do the operations to figure out the value of \begin{align*}c\end{align*}.

**Plan:** Do the multiplication and division first from left to right. Then do the addition.

**Solve:** First do the multiplication and division: \begin{align*}6 \times 7 \div 6 + 8=42 \div 6+8=7+8\end{align*}

- Last do the addition: \begin{align*}7+8=15\end{align*}
- \begin{align*}c=15\end{align*}

**Check:** \begin{align*}6 \times 7 \div 6 + 8=42 \div 6 +8=7+8=15\end{align*}

#### Earlier Problem Revisited

\begin{align*}(3 + 5) \div 2 + 4 \times 6 = a\end{align*}

We can use the problem solving steps to help.

**Describe:** The equation has parenthesis, division, multiplication and addition.

**My Job:** Do the operations to figure out the value of \begin{align*}a\end{align*}.

**Plan:** Do the parenthesis first. Then, do the multiplication and division from left to right. Finally do the addition.

**Solve:** First do the parenthesis: \begin{align*}(3 + 5) \div 2 + 4 \times 6=8 \div 2 +4 \times 6\end{align*}

- Next do the multiplication and division from left to right: \begin{align*}8 \div 2 +4 \times 6=4+24\end{align*}
- Last do the addition: \begin{align*}4+24=28\end{align*}
- \begin{align*}a=28\end{align*}

**Check:** \begin{align*}(3 + 5) \div 2 + 4 \times 6=8 \div 2 +4 \times 6 = 4+24=28\end{align*}

### Vocabulary

** Order of operations** tell us the correct order of evaluating math expressions. We always do

**Parenthesis**first. Then we do

**and finally**

*multiplication and division (from left to right)***.**

*addition and subtraction (from left to right)*### Examples

Figure out the value of the variables. Follow the order of operations.

#### Example 1

\begin{align*}(10 - 2) \div 2 + 5 \times 3 = b\end{align*}

\begin{align*}b = 19\end{align*}

#### Example 2

\begin{align*}4 + (9 + 6) \div 3 = d\end{align*}

\begin{align*}d = 9\end{align*}

#### Example 3

\begin{align*}12 + 2 \times 3 \div 6 + (7 - 4) = q\end{align*}

\begin{align*}q = 16\end{align*}

### Review

Figure out the value of the variables. Follow the order of operations.

- \begin{align*}4 \times (6 - 2) \div 2 \times 3 = t\end{align*}
- \begin{align*}2 \times (11 - 1) \div 2 = g\end{align*}
- \begin{align*}3 \times (12 - 4) - (5 - 2) = m\end{align*}
- \begin{align*}6 \times (7 + 3) \div 4 \times 2 = n\end{align*}
- \begin{align*}3 \times (5-2) - (3 -2) = p\end{align*}
- \begin{align*}2 \times (6+1) \div 2 \times 3 = q\end{align*}
- \begin{align*}5 \times (4-1) + 3 \times 8 = r\end{align*}
- \begin{align*}8 \times (5-4) \div 2 \times 5 = s\end{align*}
- \begin{align*}7 \times (7 - 3) \div 2 + 1 = u\end{align*}
- \begin{align*}5 \times (3+4) + 2 \times 3 = w\end{align*}

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### Image Attributions

Students follow the order of operations to solve for the value of unknowns. Students use problem solving steps to help.

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