# 9.3: Hanging Scales 7

**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 practice working with equations that represent what we see on scales. We will then practice solving these systems of equations.

### Guidance

In order to solve the problem above, use the problem solving steps.

- Start by
**describing**what information is given. - Then, identify what
**your job**is. In these problems, your job will be to figure out the value of each of the four variables. - Next, make a
**plan**for how you will solve. In these problems, write equations to represent the scales first. Then, solve the system of equations. - Then,
**solve**the problem. - Finally,
**check**your solution. Make sure that your solution causes each scale to have the correct weight.

#### Example A

Write equations. Figure out the weights of the blocks. Same blocks weigh the same.

**Solution:**

We can use problem solving steps to help us to write equations and find the weights of the blocks.

\begin{align*}& \mathbf{Describe}&& \text{There are four scales with blocks}.\\ &&& A: \ w, x, y, z \ \text{blocks weigh} \ 14 \ \text{pounds}.\\ &&& B: \ w, w, x \ \text{blocks weigh} \ 10 \ \text{pounds}.\\ &&& C: \ w, x, y \ \text{blocks weigh} \ 9 \ \text{pounds}.\\ &&& D: \ w, y, w, y \ \text{blocks weigh} \ 10 \ \text{pounds}.\\ \\ &\mathbf{My \ Job}&& \text{Use the scales as clues. Figure out the weights of the blocks}.\\ \\ &\mathbf{Plan} && \text{Write equations, one for each scale}.\\ &&& A: w+x+y+z=14; \ B: w+w+x=10;\\ &&& C: w+x+y=9; \ D: w+y+w+y=10.\\ &&& \text{Look for equations that are related}.\\ \\ &\mathbf{Solve}&& D: w+y+w+y=10. \ \text{There are two of each block, so} \ w+y=5\\ &&& C: w+x+y=9. \ \text{Replace} \ w+y \ \text{with}\ 5.\\ &&& \qquad 5+x=9, \ \text{and}\\ &&& \qquad x =9-5, \ \text{or} \ 4 \ pounds.\\ &&& B: w+w+x=10. \ \text{Replace} \ x \ \text{with} \ 4.\\ &&& \qquad w+w+4=10, \ 2w=6, \ \text{so} \ w=3 \ pounds.\\ &&& C: w+x+y=9. \ \text{Replace} \ x \ \text{and} \ w \ \text{with their values}. \ \text{Then} \ 3+4+y=9.\\ &&& \qquad y=9-7, \ \text{or} \ 2 \ pounds.\\ &&& A: w+x+y+z=14. \ \text{Replace known letters with their values} \\ &&& \qquad 3+4+2+z=14, \ \text{and} \ z=14-9, \ \text{or} \ 5 \ pounds.\\ \\ &\mathbf{Check} && \text{Replace each block with its weight. Check that the totals equal the number}\\ &&&\text{of pounds shown on the scales.}\\ &&& A: 3+4+2+5=14; \ B: 3+3+4=10; \ C: 3+4+2=9;\\ &&& D: 3+2+3+2=10.\end{align*}

#### Example B

Write equations. Figure out the weights of the blocks. Same blocks weigh the same.

**Solution:**

We can use problem solving steps to help us to write equations and find the weights of the blocks.

\begin{align*}& \mathbf{Describe}&& \text{There are four scales with blocks}.\\ &&& A: \ w, x, y \ \text{blocks weigh} \ 16 \ \text{pounds}.\\ &&& B: \ w, x, y, z \ \text{blocks weigh} \ 24 \ \text{pounds}.\\ &&& C: \ w, x, w, x \ \text{blocks weigh} \ 24 \ \text{pounds}.\\ &&& D: \ x, z, z \ \text{blocks weigh} \ 21 \ \text{pounds}.\\ \\ &\mathbf{My \ Job}&& \text{Use the scales as clues. Figure out the weights of the blocks}.\\ \\ &\mathbf{Plan} && \text{Write equations, one for each scale}.\\ &&& A: w+x+y=16; \ B: w + x + y + z = 24;\\ &&& C: w + x + w + x = 24; \ D: x + z + z = 21.\\ &&& \text{Look for equations that are related}.\\ \\ &\mathbf{Solve}&& C: w+x+w+x=24. \ \text{There are two of each block, so} \ w+x=12\\ &&& A: w+x+y=16. \ \text{Replace} \ w+x \ \text{with}\ 12.\\ &&& \qquad 12+y=16, \ \text{and}\\ &&& \qquad y =16-12, \ \text{or} \ 4 \ pounds.\\ &&& B: w+x+y+z=24. \ \text{Replace} \ w+x \ \text{with} \ 12 \ \text{and replace} \ y \ \text{with}\ 4.\\ &&& \qquad 12+4+z=24, \ z+16=24, \ \text{so} \ z=8 \ pounds.\\ &&& D: x+z+z=21. \ \text{Replace} \ z \ \text{with}\ 8. \ \text{Then} \ x+8+8=21.\\ &&& \qquad x=21-16, \ \text{or} \ 5 \ pounds.\\ &&& A: w+x+y=16. \ \text{Replace} \ y \ \text{with} \ 4 \ \text{and replace} \ x \ \text{with} \ 75;\\ &&& \qquad w+5+4=16, \ \text{and} \ w=16-9, \ \text{or} \ 7 \ pounds.\\ \\ &\mathbf{Check} && \text{Replace each block with its weight. Check that the totals equal the number}\\ &&&\text{of pounds shown on the scales.}\\ &&& A: 7+5+4=16; \ B: 7+5+4+8=24; \ C: 7+5+7+5=24;\\ &&& D: 5+8+8=21.\end{align*}

#### Example C

Write equations. Figure out the weights of the blocks. Same blocks weigh the same.

**Solution:**

We can use problem solving steps to help us to write equations and find the weights of the blocks.

\begin{align*}& \mathbf{Describe}&& \text{There are four scales with blocks}.\\ &&& A: \ y, z, y, z \ \text{blocks weigh} \ 28 \ \text{pounds}.\\ &&& B: \ x, y, z \ \text{blocks weigh} \ 21 \ \text{pounds}.\\ &&& C: \ w, w, y \ \text{blocks weigh} \ 18 \ \text{pounds}.\\ &&& D: \ w, x, y, z \ \text{blocks weigh} \ 26 \ \text{pounds}.\\ \\ &\mathbf{My \ Job}&& \text{Use the scales as clues. Figure out the weights of the blocks}.\\ \\ &\mathbf{Plan} && \text{Write equations, one for each scale}.\\ &&& A: y+z+y+z=28; \ B: x+y+z=21;\\ &&& C: w+w+y=18; \ D: w+x+y+z.\\ &&& \text{Look for equations that are related}.\\ \\ &\mathbf{Solve}&& A: y+z+y+z=28. \ \text{There are two of each block, so} \ y+z=14\\ &&& B: x+ y+ z = 21. \ \text{Replace} \ y+ z \ \text{with}\ 14.\\ &&& \qquad x+14=21, \ \text{and}\\ &&& \qquad x =21 - 14, \ \text{or} \ 7 \ pounds.\\ &&& D: w+x+y+z=26. \ \text{Replace} \ x \ \text{with} \ 7 \ \text{and replace} \ y+ z \ \text{with}\ 14.\\ &&& \qquad w+7+14=26, \ w+21=26, \ \text{so} \ w=5 \ pounds.\\ &&& C: w+w+y=18. \ \text{Replace} \ w \ \text{with}\ 5. \ \text{Then} \ 5+5+y=18.\\ &&& \qquad y=18-10, \ \text{or} \ 8 \ pounds.\\ &&& B: x+y+z=21. \ \text{Replace} \ y \ \text{with} \ 8 \ \text{and replace} \ x \ \text{with} \ 7;\\ &&& \qquad 7+8+z=21, \ \text{and} \ z=21-15, \ \text{or} \ 6 \ pounds.\\ \\ &\mathbf{Check} && \text{Replace each block with its weight. Check that the totals equal the number}\\ &&&\text{of pounds shown on the scales.}\\ &&& A: 8+6+8+6=28; \ B: 7+8+6=21; \ C: 5+5+8=18;\\ &&& D: 7+5+8+6=26.\end{align*}

#### Concept Problem Revisited

We can use problem solving steps to help us to write equations and find the weights of the blocks.

\begin{align*}& \mathbf{Describe}&& \text{There are four scales with blocks}.\\ &&& A: \ y, x, y, x \ \text{blocks weigh} \ 28 \ \text{pounds}.\\ &&& B: \ x, y, z \ \text{blocks weigh} \ 23 \ \text{pounds}.\\ &&& C: \ w, x, y, z \ \text{blocks weigh} \ 28 \ \text{pounds}.\\ &&& D: \ x, x, z \ \text{blocks weigh} \ 25 \ \text{pounds}.\\ \\ &\mathbf{My \ Job}&& \text{Use the scales as clues. Figure out the weights of the blocks}.\\ \\ &\mathbf{Plan} && \text{Write equations, one for each scale}.\\ &&& A: y+ x+ y+ x= 28; \ B: x+ y+ z= 23;\\ &&& C: w+ x+ y + z = 28; \ D: x+x+z=25.\\ &&& \text{Look for equations that are related}.\\ \\ &\mathbf{Solve}&& A: y+x+y+x=28. \ \text{There are two of each block, so} \ y+x=14\\ &&& B: x+ y+ z = 23. \ \text{Replace} \ x+ y \ \text{with}\ 14.\\ &&& \qquad 14 +z=23, \ \text{and}\\ &&& \qquad z =23 - 14, \ \text{or} \ 9 \ pounds.\\ &&& D: x+ x+ z = 25. \ \text{Replace} \ z \ \text{with} \ 9.\\ &&& \qquad x+x+9=25, \ 2x=25-9, \ \text{or} \ 16, \ \text{and}\\ &&& \qquad x =16 \div 2, \ \text{or}\ 8 \ pounds.\\ &&& A: x+ y= 14. \ \text{Replace} \ x \ \text{with}\ 8. \ \text{Then} \ 8+y=14.\\ &&& \qquad y=14-8, \ \text{or} \ 6 \ pounds.\\ &&& C: w+x+ y+ z = 28. \ \text{From} \ B, \ x+y+z=23. \ \text{Replace} \ x+ y+z \ \text{with} \ 23;\\ &&& \qquad w +23=28, \ \text{and} \ w=28-23, \ \text{or} \ 5 \ pounds.\\ \\ &\mathbf{Check} && \text{Replace each block with its weight. Check that the totals equal the number}\\ &&&\text{of pounds shown on the scales.}\\ &&& A: 6 + 8 + 6 + 8 =28; \ B: 8 + 6 + 9 =23; \ C: 5 + 8 + 6+9 =28;\\ &&& D: 8+8+9=25.\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

**. An**

*unknowns***is a math sentence that tells us two quantities that are equal. In this concept, we wrote**

*equation***with unknowns to represent what we saw on the scales. A**

*equations***is a set of equations that represent a given problem. Since we wrote multiple equations for each problem in this concept, we wrote a**

*system of equations***for each problem.**

*system of equations*### Guided Practice

Write equations. Figure out the weights of the blocks.

1.

2.

3.

**Answers:**

1. A: \begin{align*}x + x + z = 26\end{align*}; B: \begin{align*}w + x + z = 23\end{align*}; C: \begin{align*}x + z + x + z = 34\end{align*}; D: \begin{align*}w + y + y = 20\end{align*}.

- \begin{align*}w = 6\end{align*} pounds, \begin{align*}x = 9\end{align*} pounds, \begin{align*}y = 7\end{align*} pounds, \begin{align*}z = 8\end{align*} pounds.

2. A: \begin{align*}y + x + y + x = 26\end{align*}; B: \begin{align*}x + y + w = 17\end{align*}; C: \begin{align*}w + x + x = 20\end{align*}; D: \begin{align*}w + x + y + z = 26\end{align*}.

- \begin{align*}w = 4\end{align*} pounds, \begin{align*}x = 8\end{align*} pounds, \begin{align*}y = 5\end{align*} pounds, \begin{align*}z = 9\end{align*} pounds.

3. A: \begin{align*}w + y + z = 22\end{align*}; B: \begin{align*}x + y + z + z = 25\end{align*}; C: \begin{align*}x + y + z = 20\end{align*}; D: \begin{align*}w + z + w + z = 26\end{align*}.

- \begin{align*}w = 8\end{align*} pounds, \begin{align*}x = 6\end{align*} pounds, \begin{align*}y = 9\end{align*} pounds, \begin{align*}z = 5\end{align*} pounds.

### Practice

Write equations. Figure out the weights of the blocks. Same blocks weigh the same.

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### Image Attributions

Students determine the individual values of four different variables given information about the multiples or combinations of the variables on pictures of scales. Students use problem solving steps to help.