7.2: Distributive Property and Solving for Unknowns
Can you figure out the value of b in the equation below? Do you know how to use the distributive property and the order of operations? In this concept, you will learn how to correctly use the distributive property and order of operations to solve for unknowns.
\begin{align*}3 \times (6 + b) = 30\end{align*}
Guidance
The order of operations tells us the correct order of evaluating math expressions. We always do parentheses first. Then we do multiplication and division (from left to right) and finally addition and subtraction (from left to right). The distributive property allows us to remove parentheses when there is an unknown inside of them.
In order to evaluate expressions using the distributive property and 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 distributive property and the order of operations.
 Fourth, solve.
 Fifth, check. Substitute your answer into the equation and make sure it works.
Example A
Figure out the value of the variable. Show the steps.
\begin{align*}5 \times (a + 4) = 40\end{align*}
Solution:
We can use the problem solving steps to help.
Describe: The equation has parentheses, multiplication and addition.
My Job: Do the operations and use the distributive property to figure out the value of \begin{align*}a\end{align*}
Plan: Do the distributive property first. Then solve the equation for \begin{align*}a\end{align*}
Solve: First do the distributive property:
\begin{align*}5 \times (a + 4) &= 40\\ 5a+20 &=40\end{align*}
Next solve the equation:
\begin{align*}5a+20&=40 \\ 5a&=20\\ a&=4\end{align*}
Check:
\begin{align*}5 \times (4 + 4) &= 40\\ 5 \times 8 &= 40\\ 40&=40\end{align*}
Example B
Figure out the value of the variable. Show the steps.
\begin{align*}6 \times (c + 2) = 24\end{align*}
Solution:
We can use the problem solving steps to help.
Describe: The equation has parentheses, multiplication and addition.
My Job: Do the operations and use the distributive property to figure out the value of \begin{align*}a\end{align*}
Plan: Do the distributive property first. Then solve the equation for \begin{align*}c\end{align*}
Solve: First do the distributive property:
\begin{align*}6 \times (c + 2) &= 24\\ 6c+12 &=24\end{align*}
Next solve the equation:
\begin{align*}6c+12 &=24 \\ 6c&=12\\ c&=2\end{align*}
Check:
\begin{align*}6 \times (2 + 2) &= 24\\ 6 \times 4 &=24\\ 24&=24\end{align*}
Example C
Figure out the value of the variable. Show the steps.
\begin{align*}3 \times (3 + d) = 36\end{align*}
Solution:
We can use the problem solving steps to help.
Describe: The equation has parentheses, multiplication and addition.
My Job: Do the operations and use the distributive property to figure out the value of \begin{align*}a\end{align*}
Plan: Do the distributive property first. Then solve the equation for \begin{align*}d\end{align*}
Solve: First do the distributive property:
\begin{align*}3 \times (3 + d) &= 36\\ 9+3d &=36\end{align*}
Next solve the equation:
\begin{align*}9+3d &=36 \\ 3d&=27\\ d&=9\end{align*}
Check:
\begin{align*} 3 \times (3 + 9) &= 36\\ 3 \times 12 &=36\\ 36&=36\end{align*}
Concept Problem Revisited
\begin{align*} 3 \times (6 + b) = 30\end{align*}
We can use the problem solving steps to help.
Describe: The equation has parentheses, multiplication and addition.
My Job: Do the operations and use the distributive property to figure out the value of \begin{align*}a\end{align*}
Plan: Do the distributive property first. Then solve the equation for \begin{align*}b\end{align*}
Solve: First do the distributive property:
\begin{align*}3 \times (6 + b) &= 30\\ 18+3b &=30\end{align*}
Next solve the equation:
\begin{align*}18+3b &=30 \\ 3b&=12\\ b&=4\end{align*}
Check:
\begin{align*}3 \times (6 + 4) &= 30\\ 3 \times 10 &=30\\ 30&=30\end{align*}
Vocabulary
The order of operations tells us the correct order of evaluating math expressions. We always do parentheses first. Then we do multiplication and division (from left to right) and finally addition and subtraction (from left to right). The distributive property \begin{align*}([a \times (b + c) = a \times b + a \times c])\end{align*}
Guided Practice
Figure out the value of each variable. Show the steps.
1. \begin{align*}2 \times (4 + e) + 7 \times 4 = 46\end{align*}
2. \begin{align*}6 \div 2 \div 3 \times (f + 2) = 11\end{align*}
3. \begin{align*}37 = 3(m + 6) + 4(95)\end{align*}
Answers:
1. Here are the steps to solve:
\begin{align*}2 \times (4 + e) + 7 \times 4 &= 46\\
8 + 2e + 28 &= 46\\
36 + 2e &= 46\\
2 e &= 10\\
e &= 5\end{align*}
2. Here are the steps to solve:
\begin{align*}6 \div 2 \div 3 \times (f + 2) &= 11\\
1 \times (f + 2) &= 11\\
f + 2 &= 11\\
f & = 9\end{align*}
3. Here are the steps to solve:
\begin{align*}37 &= 3(m + 6) + 4(95)\\
37 &= 3m + 18 + 4(9  5)\\
37 &= 3m + 18 + 4 \times 4\\
37 &= 3m + 18 + 16\\
37 &= 3m + 34\\
3 &= 3m\\
1 &= m\end{align*}
Practice
Figure out the value of each variable. Show the steps.

\begin{align*} (8 + 2) \div 5 \times (g + 4) = 32\end{align*}
(8+2)÷5×(g+4)=32 
\begin{align*}70 = 5 \times (9+ t) + 4 \times (t + 4)\end{align*}
70=5×(9+t)+4×(t+4) 
\begin{align*}12 \div 6 \div 2 \times (x +9)=11\end{align*}
12÷6÷2×(x+9)=11 
\begin{align*}(2+3) \div 5 \times (n+1)=8\end{align*}
(2+3)÷5×(n+1)=8 
\begin{align*}25=5\times (u+1)+3 \times (u+4)\end{align*}
25=5×(u+1)+3×(u+4)
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Students apply the distributive property and use the order of operations to find the values of unknowns. Students use problem solving steps to help.
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