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6.5: Equations with Letters

Difficulty Level: At Grade Created by: CK-12

Look at the pictures of the scales below. Can you write equations to represent what you see on each scale? Can you figure out the value of each letter? In this concept, we will learn how to work with equations that represent what we see on scales. We will then learn how to solve sets of equations without scales.

Guidance

In order to solve the problem above, we can write equations to represent what we see on each scale. We know that if we add the weights of each of the blocks on one scale together, the total weight must be the same as the number on the scale.

In order to figure out the values of the letters, go one letter at a time. First, figure out the value of x. Then, use that information to help find the value of y. Finally, once we know x and y figure out the value of z.

Example A

Write equations. Figure out the weights of the blocks.

Solution:

We can use the problem solving steps to help.

\begin{align*}& \mathbf{Describe} && \text{There are three scales with blocks. }\!\\ &&& \text{Scale} \ D \ \text{has a} \ 5 \ \text{pound weight and one} \ y \ \text{block. They weigh} \ 7 \ \text{pounds.}\!\\ &&& \text{Scale} \ E \ \text{has 1} \ y \ \text{block and} \ 1 \ z \ \text{block. They weigh} \ 6 \ \text{pounds.}\!\\ &&& \text{Scale} \ F \ \text{has 2} \ z \ \text{blocks and 1} \ x \ \text{block. They weigh} \ 13 \ \text{pounds.}\!\\ & \mathbf{My \ Job} && \text{Figure out the weights of the blocks.}\!\\ & \mathbf{Plan} && \text{Write an equation for each scale.}\!\\ &&& \text{Scale} \ D: 5+y=7.\!\\ &&& \text{Scale} \ E: y+z=6.\!\\ &&& \text{Scale} \ F: z+z+x=13.\!\\ & \mathbf{Solve} && D: 5+y=7. \ \text{Then} \ y = 7 - 5, \ \text{or} \ 2 \ \text{pounds.}\!\\ &&& E: y + z= 6. \ \text{Replace each} \ y \ \text{with its value so} \ 2 + z = 6, \ \text{and} \ z = 4 \ \text{pounds.}\!\\ &&& F: z+z+x=13. \ \text{Replace each} \ z \ \text{with the value,}\!\\ &&& \text{so} \ 4+ 4 + x = 13, \ \text{and} \ x = 5 \ \text{pounds.}\!\\ & \mathbf{Check} && \text{Replace each block with its weight.}\!\\ &&& \text{Scale} \ D: 5+2=7 \ \text{Scale} \ E: 2+4=6 \ \text{Scale} \ F: 4+4+5=13.\end{align*}

Example B

Write equations. Figure out the weights of the blocks.

Solution:

We can use the problem solving steps to help.

\begin{align*}& \mathbf{Describe} && \text{There are three scales with blocks. }\!\\ &&& \text{Scale} \ G \ \text{has a} \ 10 \ \text{pound weight and one} \ z \ \text{block. They weigh} \ 15 \ \text{pounds.}\!\\ &&& \text{Scale} \ H \ \text{has two} \ z \ \text{blocks and} \ 1 \ x \ \text{block. They weigh} \ 17 \ \text{pounds.}\!\\ &&& \text{Scale} \ I \ \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} \ G: 10+z=15.\!\\ &&& \text{Scale} \ H: z+z+x=17.\!\\ &&& \text{Scale} \ I: y+x+z=15.\!\\ & \mathbf{Solve} && G: 10+z=15. \ \text{Then} \ z = 15 - 10, \ \text{or} \ 5 \ \text{pounds.}\!\\ &&& H: z+z+x=17. \ \text{Replace each} \ z \ \text{with its value so} \ 5+5+x=17, \ \text{and} \ x = 7 \ \text{pounds.}\!\\ &&& I: y+x+z=15. \ \text{Replace each} \ x \ \text{and} \ z \ \text{with the value,}\!\\ &&& \text{so} \ y+7+5=15, \ \text{and} \ y = 3 \ \text{pounds.}\!\\ & \mathbf{Check} && \text{Replace each block with its weight.}\!\\ &&& \text{Scale} \ G: 10+5=15 \ \text{Scale} \ H: 5+5+7=17 \ \text{Scale} \ I: 3+7+5=15.\end{align*}

Example C

Eric wrote these equations from pictures of blocks on scales. Use Eric’s equations. Find the weight of each block.

\begin{align*}\quad 5 + x = 12\!\\ {\;} \quad \ x + y = 11\!\\ {\;} \quad \ x + z = 13\end{align*}

Solution:

We can use the problem solving steps to help.

\begin{align*}& \mathbf{Describe} && \text{There are three equations with x, y, and z. }\!\\ & \mathbf{My \ Job} && \text{Figure out the values of x, y, and z.}\!\\ & \mathbf{Plan} && \text{Use the first equation to solve for x. Then, solve for y and finally solve for z.}\!\\ & \mathbf{Solve} && \text{First equation}: 5+x=12. \ \text{Then} \ x = 12 - 5, \ \text{or} \ 7.\!\\ &&& \text{Second equation}: x+y=11. \ \text{Replace each} \ x \ \text{with its value so} \ 7+y=11, \ \text{and} \ y = 4.\!\\ &&& \text{Third equation}: x + z = 13. \ \text{Replace each} \ x \ \text{with its value so} \ 7 + z = 13, \ \text{and} \ z = 6.\!\\ & \mathbf{Check} && \text{Replace each letter with its value.}\!\\ &&& \text{Scale} \ \text{First equation}: 5+7=12 \ \text{Second Equation}: 7+4=11 \ \text{Third Equation}: 7+6=13.\end{align*}

Concept Problem Revisited

We can use the problem solving steps to help.

\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*}

Vocabulary

In math, an unknown is a letter that stands for a number that we do not yet know the value of. In this concept, the blocks that we did not know the weights of were unknowns. An equation is a math sentence that tells us two quantities that are equal. In this concept, we wrote equations with unknowns to represent what we saw on the scales. A system of equations is a set of equations that represents a given problem. Since we wrote multiple equations for each problem in this concept, we wrote a system of equations for each problem.

Guided Practice

1. Figure out the weight of each block:

2. Figure out the weight of each block:

3. Figure out the values of x, y, and z.

\begin{align*}\quad 7 + y = 15\!\\ {\;} \quad \ y + z = 11\!\\ {\;} \quad \ z + x = 12\end{align*}

4. Figure out the values of x, y, and z.

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

\begin{align*}1. \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*}

\begin{align*}2. \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*}

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

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

Practice

Write equations. Figure out the weights of the blocks.

Figure out the values of x, y, and z.

1. \begin{align*}1 + 2x = 9; x + y = 10; x + y + z = 18\end{align*}
2. \begin{align*}8 + 2z = 20; z + x = 9; x + y + z = 20\end{align*}
3. \begin{align*}2y + 3 = 11; y + z = 13; 2x + z = 19\end{align*}

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