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6.12: Equations with Letters

Difficulty Level: At Grade Created by: CK-12
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Equations with Letters - Write and Solve Equations

Teacher Notes

Students are presented with three scales, their contents, and their weights. They represent relationships shown in the scales by writing equations using letters to represent unknown weights of blocks. In the next section, they solve sets or systems of equations without pictures. It is in this section that students represent sets of the same type of block as a product rather than as a sum (e.g., \begin{align*}3n\end{align*}3n instead of \begin{align*}n + n + n\end{align*}n+n+n).

Solutions

\begin{align*}1. \quad 5 + y = 7; \ y + z = 6; \ z + z + x = 13\!\\ {\;} \quad \ x = 5 \ \text{pounds}; \ y = 2 \ \text{pounds}; \ z = 4 \ \text{pounds}\end{align*}1.5+y=7; y+z=6; z+z+x=13 x=5 pounds; y=2 pounds; z=4 pounds

\begin{align*}2. \quad 10 + z = 15; \ x + z + z = 17; \ x + y + z = 15\!\\ {\;} \quad \ x = 7 \ \text{pounds}; \ y = 3 \ \text{pounds}; \ z = 5 \ \text{pounds}\end{align*}2.10+z=15; x+z+z=17; x+y+z=15 x=7 pounds; y=3 pounds; z=5 pounds

\begin{align*}3. \quad 7 + x = 13; \ x + x + y = 16; \ x + y + z = 17\!\\ {\;} \quad \ x = 6 \ \text{pounds}; \ y = 4 \ \text{pounds}; \ z = 9 \ \text{pounds}\end{align*}3.7+x=13; x+x+y=16; x+y+z=17 x=6 pounds; y=4 pounds; z=9 pounds

\begin{align*}4. \quad x + x + y = 9; \ y + y + z = 13; \ 9 + z = 12\!\\ {\;} \quad \ x = 2 \ \text{pounds}; \ y = 5 \ \text{pounds}; \ z = 3 \ \text{pounds}\end{align*}4.x+x+y=9; y+y+z=13; 9+z=12 x=2 pounds; y=5 pounds; z=3 pounds

\begin{align*}5. \quad x + z + z = 18; \ 6 + z = 14; \ x + y + z = 17\!\\ {\;} \quad \ x = 2 \ \text{pounds}; \ y = 7 \ \text{pounds}; \ z = 8 \ \text{pounds}\end{align*}5.x+z+z=18; 6+z=14; x+y+z=17 x=2 pounds; y=7 pounds; z=8 pounds

\begin{align*}6. \quad 4 + y + y = 16; \ x + y = 11; \ x + y + z = 19\!\\ {\;} \quad \ x = 5 \ \text{pounds}; \ y = 6 \ \text{pounds}; \ z = 8 \ \text{pounds}\end{align*}6.4+y+y=16; x+y=11; x+y+z=19 x=5 pounds; y=6 pounds; z=8 pounds

\begin{align*}7. \ x = 7 \ \text{pounds}; \ y = 4 \ \text{pounds}; \ z = 6 \ \text{pounds}\end{align*}7. x=7 pounds; y=4 pounds; z=6 pounds

\begin{align*}8. \ x = 9 \ \text{pounds}; \ y = 8 \ \text{pounds}; \ z = 3 \ \text{pounds}\end{align*}

\begin{align*}9. \ x = 4 \ \text{pounds}; \ y = 3 \ \text{pounds}; \ z = 5 \ \text{pounds}\end{align*}

\begin{align*}10. \ x = 4 \ \text{pounds}; \ y = 6 \ \text{pounds}; \ z = 8 \ \text{pounds}\end{align*}

\begin{align*}11. \ x = 3 \ \text{pounds}; \ y = 11 \ \text{pounds}; \ z = 6 \ \text{pounds}\end{align*}

\begin{align*}12. \ x = 5 \ \text{pounds}; \ y = 4 \ \text{pounds}; \ z = 9 \ \text{pounds}\end{align*}

\begin{align*}& \mathbf{Describe} && \text{There are three scales with blocks. }\!\\ &&& \text{Scale} \ A \ \text{has an} \ 8 \ \text{pound weight and one} \ x \ \text{block. They weigh} \ 13 \ \text{pounds.}\!\\ &&& \text{Scale} \ B \ \text{has one} \ y \ \text{block and} \ 2 \ x \ \text{blocks. They weigh} \ 14 \ \text{pounds.}\!\\ &&& \text{Scale} \ C \ \text{has one} \ x \ \text{block, one} \ y \ \text{block, and one} \ z \ \text{block. They weigh} \ 15 \ \text{pounds.}\!\\ & \mathbf{My \ Job} && \text{Figure out the weights of the blocks.}\!\\ & \mathbf{Plan} && \text{Write an equation for each scale.}\!\\ &&& \text{Scale} \ A: 8 + x = 13\!\\ &&& \text{Scale} \ B: y + x + x = 14.\!\\ &&& \text{Scale} \ C: x + y + z = 15.\!\\ & \mathbf{Solve} && A: 8 + x = 13. \ \text{Then} \ x = 13 - 8, \ \text{or} \ 5 \ \text{pounds.}\!\\ &&& B: y + x + x = 14. \ \text{Replace each} \ x \ \text{with its value so} \ y + 5 + 5 = 14, \ \text{and} \ y = 4 \ \text{pounds.}\!\\ &&& C: x + y + z = 12. \ \text{Replace each} \ x \ \text{and} \ y \ \text{with the value,}\!\\ &&& \text{so} \ 5 + 4 + z = 15, \ \text{and} \ z = 6 \ \text{pounds.}\!\\ & \mathbf{Check} && \text{Replace each block with its weight.}\!\\ &&& \text{Scale} \ A: 8 + 5 = 13 \ \text{Scale} \ B: 4 + 5 + 5 = 14 \ \text{Scale} \ C: 5 + 4 + 6 = 15.\end{align*}

Write equations. Figure out the weights of the blocks.

Eric wrote these equations from pictures of blocks on scales.

Use Eric’s equations. Find the weight of each block.

\begin{align*}7. \quad 5 + x = 12\!\\ {\;} \quad \ x + y = 11\!\\ {\;} \quad \ x + z = 13\!\\ {\;} \quad \ x = \underline{\;\;\;\;\;\;} \ \text{pounds}\!\\ {\;} \quad \ y = \underline{\;\;\;\;\;\;} \ \text{pounds}\!\\ {\;} \quad \ z = \underline{\;\;\;\;\;\;} \ \text{pounds}\end{align*}

\begin{align*}8. \quad 7 + y = 15\!\\ {\;} \quad \ y + z = 11\!\\ {\;} \quad \ z + x = 12\!\\ {\;} \quad \ x = \underline{\;\;\;\;\;\;} \ \text{pounds}\!\\ {\;} \quad \ y = \underline{\;\;\;\;\;\;} \ \text{pounds}\!\\ {\;} \quad \ z = \underline{\;\;\;\;\;\;} \ \text{pounds}\end{align*}

\begin{align*}9. \quad z + 6 = 11\!\\ {\;} \quad \ x + z = 9\!\\ {\;} \quad \ x + y + z = 12\!\\ {\;} \quad \ x = \underline{\;\;\;\;\;\;} \ \text{pounds}\!\\ {\;} \quad \ y = \underline{\;\;\;\;\;\;} \ \text{pounds}\!\\ {\;} \quad \ z = \underline{\;\;\;\;\;\;} \ \text{pounds}\end{align*}

\begin{align*}10. \ 1 + 2x = 9\!\\ {\;} \quad \ x + y = 10\!\\ {\;} \quad \ x + y + z = 18\!\\ {\;} \quad \ x = \underline{\;\;\;\;\;\;} \ \text{pounds}\!\\ {\;} \quad \ y = \underline{\;\;\;\;\;\;} \ \text{pounds}\!\\ {\;} \quad \ z = \underline{\;\;\;\;\;\;} \ \text{pounds}\end{align*}

\begin{align*}11. \ 8 + 2z = 20\!\\ {\;} \quad \ z + x = 9\!\\ {\;} \quad \ x + y + z = 20\!\\ {\;} \quad \ x = \underline{\;\;\;\;\;\;} \ \text{pounds}\!\\ {\;} \quad \ y = \underline{\;\;\;\;\;\;} \ \text{pounds}\!\\ {\;} \quad \ z = \underline{\;\;\;\;\;\;} \ \text{pounds}\end{align*}

\begin{align*}12. \ 2y + 3 = 11\!\\ {\;} \quad \ y + z = 13\!\\ {\;} \quad \ 2x + z = 19\!\\ {\;} \quad \ x = \underline{\;\;\;\;\;\;} \ \text{pounds}\!\\ {\;} \quad \ y = \underline{\;\;\;\;\;\;} \ \text{pounds}\!\\ {\;} \quad \ z = \underline{\;\;\;\;\;\;} \ \text{pounds}\end{align*}

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