7.1: Solve for the Unknown
Can you use the order of operations? What's the value of \begin{align*}a\end{align*}
\begin{align*}(3 + 5) \div 2 + 4 \times 6 = a\end{align*}
Guidance
The order of operations tells us the correct order of evaluating math expressions. We always do parenthesis first. Then we do multiplication and division (from left to right) and finally 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.
Example A
Figure out the value of the variable. Follow the order of operations.
\begin{align*}8(31)+6\times 2=e\end{align*}
Solution:
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(31)+6\times 2=82+6\times 2\end{align*}

Next do the multiplication: \begin{align*}82+6\times 2=82+12\end{align*}
8−2+6×2=8−2+12 
Last do the subtraction and addition: \begin{align*}82+12=6+12=18\end{align*}
8−2+12=6+12=18 
\begin{align*}e=18\end{align*}
e=18
Check: \begin{align*}8(31)+6\times 2=82+6\times 2=82+12=6+12=18\end{align*}
Example B
Figure out the value of the variable. Follow the order of operations.
\begin{align*}(4 + 6) \div 5  3 \div 3 = a\end{align*}
Solution:
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 53 \div 3\end{align*}

Next do the division: \begin{align*}10\div 53 \div 3=21\end{align*}
10÷5−3÷3=2−1 
Last do the subtraction: \begin{align*}21=1\end{align*}
2−1=1 
\begin{align*}a=1\end{align*}
a=1
Check: \begin{align*}(4 + 6) \div 5  3 \div 3 =10\div 53 \div 3=21=1\end{align*}
Example C
Figure out the value of the variable. Follow the order of operations.
\begin{align*}6 \times 7 \div 6 + 8 = c\end{align*}
Solution:
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*}
7+8=15 
\begin{align*}c=15\end{align*}
c=15
Check: \begin{align*}6 \times 7 \div 6 + 8=42 \div 6 +8=7+8=15\end{align*}
Concept 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*}
8÷2+4×6=4+24 
Last do the addition: \begin{align*}4+24=28\end{align*}
4+24=28 
\begin{align*}a=28\end{align*}
a=28
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 multiplication and division (from left to right) and finally addition and subtraction (from left to right).
Guided Practice
Figure out the value of the variables. Follow the order of operations.
1. \begin{align*}(10  2) \div 2 + 5 \times 3 = b\end{align*}
2. \begin{align*}4 + (9 + 6) \div 3 = d\end{align*}
3. \begin{align*}12 + 2 \times 3 \div 6 + (7  4) = q\end{align*}
Answers:
1. \begin{align*}b = 19\end{align*}
2. \begin{align*}d = 9\end{align*}
3. \begin{align*}q = 16\end{align*}
Practice
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 (52)  (3 2) = p\end{align*}
 \begin{align*}2 \times (6+1) \div 2 \times 3 = q\end{align*}
 \begin{align*}5 \times (41) + 3 \times 8 = r\end{align*}
 \begin{align*}8 \times (54) \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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Students follow the order of operations to solve for the value of unknowns. Students use problem solving steps to help.