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6.4: Distributive Property and Solving for Unknowns

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Distributive Property and Solving for Unknowns - Apply the Distributive Property

Teacher Notes

Students apply the distributive property [a \times (b + c) = a \times b + a \times c] and the rule for the order of operations to solve for the value of the unknown in each equation. In each problem, the unknown is in the parentheses. Students show the steps performed to identify the value of the unknown.

Solutions

1. \quad 5 \times (a + 4) = 40\!\\{\;} \quad \ 5a + 20 = 40\!\\{\;} \quad \ 5a = 20\!\\{\;} \quad  \ a = 4

2. \quad 6 \times (c + 2) = 24\!\\{\;} \quad \ 6c + 12 = 24\!\\{\;} \quad \ 6c = 12\!\\{\;} \quad \ c = 2

3. \quad 3 \times (3 + d) = 36\!\\{\;} \quad \ 9 + 3d = 36\!\\{\;} \quad \ 3d = 27\!\\{\;} \quad \ d = 9

4. \quad 2 \times (4 + e) + 7 \times 4 = 46\!\\{\;} \quad \ 8 + 2e + 28 = 46\!\\{\;} \quad \ 36 + 2e = 46\!\\{\;} \quad \ 2 e = 10\!\\{\;} \quad \ e = 5

 3 \times (6 + b) = 30

\mathbf{What’s \ the \ value \ of} \ b?

& \mathbf{Describe:} && \text{The equation shows the variable} \ b, \ \text{and two operations.}\\&&& 6 + b \ \text{is in parentheses and is multiplied by} \ 3.\\& \mathbf{My \ Job:} && \text{Do the operations to figure out the value of} \ b.\\& \mathbf{Plan:} && \text{Apply the distributive property. Then solve for} \ b.\\& \mathbf{Solve:} && \text{Apply the distributive property:} \ 3 \times (6 + b) = 3 \times 6 + 3 \times b\\&&& \text{Apply the order of operations:} \ 3 \times 6 + 3 \times b = 30\\&&& \qquad \qquad \qquad \qquad \qquad \qquad \qquad 18 + 3b = 30\\&&& \qquad \qquad \qquad \qquad \qquad \qquad \qquad 3b =  12\\&&& \qquad \qquad \qquad \qquad \qquad \qquad \qquad b = 4\\& \mathbf{Check:} && \text{Replace} \ b \ \text{with 4 in the equation. Check that the expressions name the}\\&&& \text{same number.}\\&&& 3 \times (6 + b) = 30\\&&& 3 \times (6 + 4) = 30\\&&& 3 \times 10 = 30\\&&& 30 = 30

Figure out the value of each variable.

Show the steps.

1. \quad 5 \times (a + 4) = 40\!\\\!\\{\;} \quad \ a = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\!\\\!\\2. \quad 6 \times (c + 2) = 24\!\\\!\\{\;} \quad \ c = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\!\\\!\\3. \quad 3 \times (3 + d) = 36\!\\\!\\{\;} \quad \ d = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\!\\\!\\4. \quad 2 \times (4 + e) + 7 \times 4 = 46\!\\\!\\{\;} \quad \ e = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}

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