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8.4: Hanging Scales 6

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 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 three 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.

Solution:

We can use problem solving steps to help.

$& \mathbf{Describe:} && \text{There are three scales with blocks.}\\&&& \text{A: Two} \ x \ \text{and two} \ z \ \text{blocks. They weigh 20 pounds.}\\&&& \text{B: One} \ x, \ \text{one} \ y, \ \text{and one} \ z \ \text{block. They weigh 15 pounds.}\\&&& \text{C: One} \ x \ \text{and two} \ y \ \text{blocks. They weigh 14 pounds.}\\& \mathbf{My \ Job:} && \text{Use the scales as clues. Figure out the weights of the blocks.}\\& \mathbf{Plan:} && \text{Write equations, one for each scale.}\\&&& A: z+x+z+x=20; \ B: x+y+z=15; \ C:x+y+y=14\\&&& \text{Solve the equations.}\\& \mathbf{Solve:} && A: z+x+z+x=20. \ \text{There are two of each block, so} \ z+x=10\\&&& \text{B}: (z + x) +y = 15. \ \text{Replace} \ (z + x) \ \text{with} \ 10.\\&&& \quad \quad 10 + y = 15, \ \text{and}\\&&& \quad \quad y = 15 - 10, \ \text{or} \ 5 \ \text{pounds.}\\&&& C: x+y+y=14. \ \text{Replace each} \ y \ \text{with} \ 5.\\&&& \quad \quad x + 10 = 14, \ \text{and}\\&&& \quad \quad x = 14 - 10, \text{or} \ 4 \ \text{pounds}\\&&& A: z + x = 10. \ \text{Replace} \ x \ \text{with} \ 4. \ \text{Then}\ z + 4 = 10.\\&&& \quad \quad z = 10 - 4, \ \text{or} \ 6 \ \text{pounds.}\\& \mathbf{Check:} && \text{Replace each block with its weight. Check that the total equal the number of}\\&&& \text{pounds shown on the scales.} \\&&& A: 6+4+6+4=20; \ B: 4+5+6=15; \ C: 4+5+5=14.$

Example B

Write equations. Figure out the weights of the blocks.

Solution:

We can use problem solving steps to help.

$& \mathbf{Describe:} && \text{There are three scales with blocks.}\\&&& \text{D: One} \ x, \ \text{one} \ y, \ \text{and one} \ z \ \text{block. They weigh 19 pounds.}\\&&& \text{E: Two} \ y \ \text{and two} \ z \ \text{blocks. They weigh 24 pounds.}\\&&& \text{F: One} \ z \ \text{and two} \ x \ \text{blocks. They weigh 24 pounds.}\\& \mathbf{My \ Job:} && \text{Use the scales as clues. Figure out the weights of the blocks.}\\& \mathbf{Plan:} && \text{Write equations, one for each scale.}\\&&& D: x+y+z=19; \ E: y+z+y+z=24; \ F:x+x+z=24\\&&& \text{Solve the equations.}\\& \mathbf{Solve:} && E: y+z+y+z=24. \ \text{There are two of each block, so} \ y+z=12\\&&& \text{D}: x+(y+z) = 19. \ \text{Replace} \ (y+z) \ \text{with} \ 12.\\&&& \quad \quad x+12=19, \ \text{and}\\&&& \quad \quad x = 19 - 12, \ \text{or} \ 7 \ \text{pounds.}\\&&& F: x+x+z=24. \ \text{Replace each} \ x \ \text{with} \ 7.\\&&& \quad \quad 14 + z = 24, \ \text{and}\\&&& \quad \quad z = 24 - 14, \text{or} \ 10 \ \text{pounds}\\&&& E: y+z=12. \ \text{Replace} \ z \ \text{with} \ 10. \ \text{Then}\ y+10=12.\\&&& \quad \quad y = 12 - 10, \ \text{or} \ 2 \ \text{pounds.}\\& \mathbf{Check:} && \text{Replace each block with its weight. Check that the total equal the number of}\\&&& \text{pounds shown on the scales.} \\&&& D:7+2+10=19; \ E: 2+10+2+10=24; \ F: 7+7+10=24.$

Example C

Write equations. Figure out the weights of the blocks.

Solution:

We can use problem solving steps to help.

$& \mathbf{Describe:} && \text{There are three scales with blocks.}\\&&& \text{G: One} \ x, \ \text{one} \ y, \ \text{and one} \ z \ \text{block. They weigh 20 pounds.}\\&&& \text{H: Two} \ x \ \text{and two} \ y \ \text{blocks. They weigh 26 pounds.}\\&&& \text{I: One} \ y \ \text{and two} \ z \ \text{blocks. They weigh 22 pounds.}\\& \mathbf{My \ Job:} && \text{Use the scales as clues. Figure out the weights of the blocks.}\\& \mathbf{Plan:} && \text{Write equations, one for each scale.}\\&&& G:x+y+z=20; \ H: y+x+y+x=26; \ I:y+z+z=22\\&&& \text{Solve the equations.}\\& \mathbf{Solve:} && H: y+x+y+x=26. \ \text{There are two of each block, so} \ y+x = 13\\&&& \text{G}: (x+y) + z = 20. \ \text{Replace} \ (x + y) \ \text{with} \ 13.\\&&& \quad \quad 13 + z = 20, \ \text{and}\\&&& \quad \quad z = 20 - 13, \ \text{or} \ 7 \ \text{pounds.}\\&&& I: y + z + z = 22. \ \text{Replace each} \ z \ \text{with} \ 7.\\&&& \quad \quad y + 14 = 22, \ \text{and}\\&&& \quad \quad y = 22 - 14, \text{or} \ 8 \ \text{pounds}\\&&& H: y+x=13. \ \text{Replace} \ y \ \text{with} \ 8. \ \text{Then}\ 8 + x = 13.\\&&& \quad \quad x = 13 - 8, \ \text{or} \ 5 \ \text{pounds.}\\& \mathbf{Check:} && \text{Replace each block with its weight. Check that the total equal the number of}\\&&& \text{pounds shown on the scales.} \\&&& G: 5+7+8=20; \ H: 8+5+8+5=26; \ I: 8+7+7=22.$

Concept Problem Revisited

We can use problem solving steps to help.

$& \mathbf{Describe:} && \text{There are three scales with blocks.}\\&&& \text{A: Two} \ x \ \text{and two} \ y \ \text{blocks. They weigh 26 pounds.}\\&&& \text{B: One} \ x, \ \text{one} \ y, \ \text{and one} \ z \ \text{block. They weigh 22 pounds.}\\&&& \text{C: One} \ x \ \text{and two} \ z \ \text{blocks. They weigh 24 pounds.}\\& \mathbf{My \ Job:} && \text{Use the scales as clues. Figure out the weights of the blocks.}\\& \mathbf{Plan:} && \text{Write equations, one for each scale.}\\&&& A: x + y + x + y = 26; \ B: x + y + z = 22; \ C:x + z + z = 24\\&&& \text{Solve the equations.}\\& \mathbf{Solve:} && A: x + y + x + y = 26. \ \text{There are two of each block, so} \ x + y = 13\\&&& \text{B}: (x + y) + z = 22. \ \text{Replace} \ (x + y) \ \text{with} \ 13.\\&&& \quad \quad 13 + z = 22, \ \text{and}\\&&& \quad \quad z = 22 - 13, \ \text{or} \ 9 \ \text{pounds.}\\&&& C: x + z + z = 24. \ \text{Replace each} \ z \ \text{with} \ 9.\\&&& \quad \quad x + 18 = 24, \ \text{and}\\&&& \quad \quad x = 24 - 18, \text{or} \ 6 \ \text{pounds}\\&&& A: x + y = 13. \ \text{Replace} \ x \ \text{with} \ 6. \ \text{Then}\ 6 + y = 13.\\&&& \quad \quad y = 13 - 6, \ \text{or} \ 7 \ \text{pounds.}\\& \mathbf{Check:} && \text{Replace each block with its weight. Check that the total equal the number of}\\&&& \text{pounds shown on the scales.} \\&&& A: 6 + 7 + 6 + 7 = 26; \ B: 6 + 7 + 9 = 23; \ C: 6 + 9+ 9 = 24.$

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

Write equations. Figure out the weights of the blocks.

1.

2.

3.

1. J: $x + y + z = 22$ ; K: $x + x + y = 25$ ; $L: x + z + x+ z= 26$

$x= 8, \ y = 9, \ z = 5$

2. M: $y + x + y + x = 32$ ; N: $z + z + x = 28$ ; P: $x+ y+ z = 25$

$x= 10, \ y = 6, \ z = 9$

3. Q: $x + y + z = 28$ ; R: $y + z + y+ z= 34$ ; S: $x + x+ y= 32$

$x= 11, \ y = 10, \ z = 7$

Practice

Write equations. Figure out the weights of the blocks.

Jan 18, 2013