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# 8.2: Solve for Unknowns

Difficulty Level: At Grade Created by: CK-12

Look at the equation below. Can you figure out the value of \begin{align*}a\end{align*}? In this concept, we will practice using the order of operations and the distributive property to solve equations.

\begin{align*}4 \times 1 + 6(a - 1) - 9 - 2^3 = 2 + 3^2\end{align*}

### Guidance

The order of operations tells us the correct order of evaluating math expressions. We always do parentheses first. Then we do exponents. Next 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*} 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

Solve for the unknown. Follow the order of operations and show each step.

\begin{align*}(3 + 5)^2 \div 2 - 2 = 2b + 4^2\end{align*}

Solution:

We can use the problem solving steps to help us with the order of operations.

\begin{align*}& \mathbf{Describe:} && \text{The} \ 3+5 \ \text{is in parentheses and has an exponent of 2. The 4 has an exponent of 2.}\\ &&&\text{The other operations shown are multiplication, division, subtraction and addition.}\\ & \mathbf{My \ Job:} && \text{Do the order of operations in order to solve for the value of} \ d.\\ & \mathbf{Solve:} && (3 + 5)^2 \div 2 - 2 = 2b + 4^2\\ &&& \mathbf{Parentheses} \qquad \qquad 8^2 \div 2 - 2 = 2b + 4^2\\ &&& \mathbf{Exponents} \qquad \qquad \quad 64 \div 2 - 2 = 2b + 16\\ &&& \mathbf{Multiplication/} \qquad \ \ 32 - 2 = 2b + 16\\ &&& \mathbf{Division}\\ &&& \mathbf{(left \ to \ right)}\\ &&& \mathbf{Addition} \qquad \qquad \qquad 30 = 2b + 16\\ &&& \mathbf{(left \ to \ right)} \qquad \qquad 2b= 30-16\\ &&& \qquad \qquad \qquad \qquad \qquad \ \ 2b = 14\\ &&& \qquad \qquad \qquad \qquad \qquad \ \ b = 7\\ & \mathbf{Check:} && \text{Replace} \ b \ \text{with 7 in the equation. Check that the two expressions (to the right and}\\ &&&\text{to the left of the = symbol) name the same number.}\\ &&& (3 + 5)^2 \div 2 - 2 = 2\times 7 + 4^2\\ &&& 8^2 \div 2 - 2 = 2\times 7 + 4^2\\ &&& 64 \div 2 - 2 = 2\times 7 +16\\ &&& 32 - 2 = 14 +16\\ &&& 30=30\end{align*}

#### Example B

Solve for the unknown. Follow the order of operations and show each step.

\begin{align*}4(c + 2) + 2 = 2 \times 5^2\end{align*}

Solution:

We can use the problem solving steps to help us with the order of operations.

\begin{align*}& \mathbf{Describe:} && \text{The} \ c+2 \ \text{is in parentheses and is multiplied by 4. The 5 has an exponent of 2.}\\ &&&\text{The other operations shown are multiplication and addition.}\\ & \mathbf{My \ Job:} && \text{Apply the distributive property and do the order of operations in order to solve for}\\ &&&\text{the value of} \ c.\\ & \mathbf{Solve:} && 4(c + 2) + 2 = 2 \times 5^2\\ &&& \mathbf{Distributive} \qquad \qquad 4\times c + 4\times 2 + 2 = 2 \times 5^2\\ &&& \mathbf{property}\\ &&& \mathbf{Exponents} \qquad \qquad \quad 4\times c + 4\times 2 + 2 = 2 \times 25\\ &&& \mathbf{Multiplication/} \qquad \ \ 4c + 8 + 2 = 50\\ &&& \mathbf{Division} \\ &&& \mathbf{(left \ to \ right)}\\ &&& \mathbf{Addition} \qquad \qquad \qquad 4c + 10 = 50\\ &&& \mathbf{(left \ to \ right)} \qquad \qquad 4c = 50-10\\ &&& \qquad \qquad \qquad \qquad \qquad \ \ 4c = 40\\ &&& \qquad \qquad \qquad \qquad \qquad \ \ c = 10\\ & \mathbf{Check:} && \text{Replace} \ c \ \text{with 10 in the equation. Check that the two expressions (to the right and}\\ &&&\text{to the left of the = symbol) name the same number.}\\ &&& 4(10 + 2) + 2 = 2 \times 5^2\\ &&& 4\times 12 + 2 = 2 \times 5^2\\ &&& 4\times 12 + 2 = 2 \times 25\\ &&& 48 + 2 = 20\\ &&& 50 = 50\end{align*}

#### Example C

Solve for the unknown. Follow the order of operations and show each step.

\begin{align*}2 + (16 - 4) \div 3 + d + 2d = 9^2 \div 9 \times 2\end{align*}

Solution:

We can use the problem solving steps to help us with the order of operations.

\begin{align*}& \mathbf{Describe:} && \text{The} \ 16-4 \ \text{is in parentheses. The 9 has an exponent of 2.}\\ &&&\text{The other operations shown are multiplication, division, and addition.}\\ & \mathbf{My \ Job:} && \text{Do the order of operations in order to solve for}\\ &&&\text{the value of} \ d.\\ & \mathbf{Solve:} && 2 + (16 - 4) \div 3 + d + 2d = 9^2 \div 9 \times 2\\ &&& \mathbf{Parentheses} \qquad \qquad 2 + 12 \div 3 + d + 2d = 9^2 \div 9 \times 2\\ &&& \mathbf{Exponents} \qquad \qquad \quad 2 + 12 \div 3 + d + 2d = 81 \div 9 \times 2\\ &&& \mathbf{Multiplication/} \qquad \ \ 2 + 4 + d + 2d = 9 \times 2\\ &&& \mathbf{Division} \qquad \qquad \qquad \ \ 2 + 4 + d + 2d = 18\\ &&& \mathbf{(left \ to \ right)}\\ &&& \mathbf{Addition} \qquad \qquad \qquad 6 + 3d = 18\\ &&& \mathbf{(left \ to \ right)} \qquad \qquad 3d = 18-6\\ &&& \qquad \qquad \qquad \qquad \qquad \ \ 3d = 12\\ &&& \qquad \qquad \qquad \qquad \qquad \ \ d = 4\\ & \mathbf{Check:} && \text{Replace} \ d \ \text{with 4 in the equation. Check that the two expressions (to the right and}\\ &&&\text{to the left of the = symbol) name the same number.}\\ &&& 2 + (16 - 4) \div 3 + 4 + 2\times 4 = 9^2 \div 9 \times 2\\ &&& 2 + 12 \div 3 + 4 + 2\times 4 = 9^2 \div 9 \times 2\\ &&& 2 + 12 \div 3 + 4 + 2\times 4 = 81 \div 9 \times 2\\ &&& 2 + 4 + 4 + 8 = 18\\ &&& 18 = 18\end{align*}

#### Concept Problem Revisited

\begin{align*}4 \times 1 + 6(a - 1) - 9 - 2^3 = 2 + 3^2\end{align*}

We can use the problem solving steps to help us with the order of operations.

\begin{align*}& \mathbf{Describe:} && \text{The} \ a - 1 \ \text{is in parentheses and is multiplied by 6. The 2 has an exponent of 3, and}\\ &&&\text{the 3 has an exponent of 2. The other operations shown are multiplication,}\\ &&&\text{addition, and subtraction.}\\ & \mathbf{My \ Job:} && \text{Apply the distributive property and do the order of operations in order to solve for}\\ &&&\text{the value of} \ a.\\ & \mathbf{Solve:} && 4 \times 1 + 6(a - 1) - 9 - 2^3 = 2 + 3^2\\ &&& \mathbf{Distributive} \qquad \qquad 4 \times 1 + 6a - 6 - 9 - 2^3 = 2 + 3^2\\ &&& \mathbf{property}\\ &&& \mathbf{Exponents} \qquad \qquad \quad 4 \times 1 + 6a - 6 - 9 - 8 = 2 + 9\\ &&& \mathbf{Multiplication/} \qquad \ \ 4 + 6a - 6 - 9 - 8 = 2 + 9\\ &&& \mathbf{Division}\\ &&& \mathbf{(left \ to \ right)}\\ &&& \mathbf{Addition} \qquad \qquad \qquad 6a - 19 = 11\\ &&& \mathbf{(left \ to \ right)} \qquad \qquad 6a = 11 + 19\\ &&& \qquad \qquad \qquad \qquad \qquad \ \ 6a = 30\\ &&& \qquad \qquad \qquad \qquad \qquad \ \ a = 5\\ & \mathbf{Check:} && \text{Replace} \ a \ \text{with 5 in the equation. Check that the two expressions (to the right and}\\ &&&\text{to the left of the = symbol) name the same number.}\\ &&& 4 \times 1 + 6(a - 1) - 9 - 2^3 = 2 + 3^2\\ &&& 4 \times 1 + 6(5 - 1) - 9 - 2^3 = 2 + 3^2\\ &&& 4 \times 1 + 6 \times 4 - 9 - 2^3 = 2 + 3^2\\ &&& 4 \times 1 + 6 \times 4 - 9 - 8 = 2 + 9\\ &&& 4 + 24 - 9 - 8 = 2 + 9\\ &&& 11 = 11\end{align*}

### Vocabulary

The order of operations tells us the correct order of evaluating math expressions. We always do parentheses first. Then we do exponents. Next 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*} allows us to remove parentheses when there is an unknown inside of them.

### Guided Practice

For each problem, solve for the unknown. Follow the order of operations and show each step.

1. \begin{align*}61 - (9 - 6)^2 = 5 (2 + f) - 3^1\end{align*}

2. \begin{align*}5h + 3(h - 2) + 2 = 2^2 \times 3\end{align*}

3. \begin{align*}10^2 - 5 \times 8 = 1 + 6(j + 4) + (5^2 + 5) \div 6\end{align*}

Answers:

1. Here are the steps to solve:

\begin{align*} 61 - (9 - 6)^2 &= 5(2 + f) - 3^1\\ 61 - 3^2 &= 5 \times 2 + 5f - 3^1\\ 61 - 9 &= 5 \times 2 + 5f - 3\\ 61 - 9 &= 10 + 5f - 3\\ 52 &= 5f + 7\\ 52 - 7 &= 5f\\ 45 &= 5f\\ 9 &= f\end{align*}

2. Here are the steps to solve:

\begin{align*}5h + 3(h - 2) + 2 &= 2^2 \times 3\\ 5h + 3h - 6 + 2 &= 2^2 \times 3\\ 5h + 3h - 6 + 2 &= 4 \times 3\\ 5h + 3h - 6 + 2 &= 12\\ 8h - 4 &= 12\\ 8h &= 12 + 4\\ 8h &= 16\\ h &= 2\end{align*}

3. Here are the steps to solve:

\begin{align*} 10^2 - 5 \times 8 &= 1 + 6 (j + 4) + (5^2 + 5) \div 6\\ 10^2 - 5 \times 8 &= 1 + 6 j + 24 + (25 + 5) \div 6\\ 10^2 - 5 \times 8 &= 1 + 6 j + 24 + 30 \div 6\\ 100 - 5 \times 8 &= 1 + 6 j + 24 + 30 \div 6\\ 100 - 40 &= 1 + 6 j + 24 + 5\\ 60 &= 6 j + 30\\ 60 - 30 &= 6 j\\ 30 &= 6 j\\ 5 &= j\end{align*}

### Practice

For each problem, solve for the unknown. Follow the order of operations and show each step.

1. \begin{align*}k + 2(6 + k) + 3^2 = 2^3 \times 3 \times 2 - 3\end{align*}
2. \begin{align*}6^2 + 2^2 = 2(2m + 4m) - 4^2 \times 2\end{align*}
3. \begin{align*}27-(6-3)^2=17(2+g)-4^2\end{align*}
4. \begin{align*}6f+2(f-4)+8=2^4\times 4\end{align*}
5. \begin{align*}5^3-4\times 15 = 7 +6(k+3)+(4^2+4)\div 5\end{align*}

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Date Created:
Jan 18, 2013
Last Modified:
Mar 23, 2016
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