Question

find a solution to each equation. use the hangers to explain what each solution means.

1. here are some balanced hangers. each piece is labeled with its weight.

for each diagram:

1. the following listing of quantities contains solutions to each of the hangers. determine which value of the variable would make the each hanger balanced. 2, 2.75, 3, 3.5, 4, 5, 6, 6.75, 7, 8, 9, 10
2. write an equation for each hanger.
3. explain how to reason with the diagram to find the weight of a piece with a letter.

Explanation:

Step1: Solve for $x$ (Hanger A)

Balanced weights sum to equal value.
$x + 3 = 8$
Subtract 3 from both sides:
$x = 8 - 3 = 5$

Step2: Solve for $y$ (Hanger B)

Two $y$ equal total weight.
$2y = 12$
Divide by 2 on both sides:
$y = \frac{12}{2} = 6$

Step3: Solve for $z$ (Hanger C)

Four $z$ equal total weight.
$4z = 11$
Divide by 4 on both sides:
$z = \frac{11}{4} = 2.75$

Step4: Solve for $w$ (Hanger D)

$w$ plus given weight equals total.
$w + 3\frac{1}{2} = 13\frac{1}{2}$
Subtract $3\frac{1}{2}$ from both sides:
$w = 13\frac{1}{2} - 3\frac{1}{2} = 10$

Answer:

1. Matching values:

• Hanger A: $x=5$
• Hanger B: $y=6$
• Hanger C: $z=2.75$
• Hanger D: $w=10$

2. Equations:

• Hanger A: $x + 3 = 8$
• Hanger B: $2y = 12$
• Hanger C: $4z = 11$
• Hanger D: $w + 3.5 = 13.5$

3. Reasoning:

• Hanger A: The total weight on the left equals the right. Subtract the known 3 from 8 to find $x$.
• Hanger B: Two identical $y$ pieces balance 12, so divide 12 by 2 to find one $y$.
• Hanger C: Four identical $z$ pieces balance 11, so divide 11 by 4 to find one $z$.
• Hanger D: The total weight on the right equals the left. Subtract the known $3.5$ from $13.5$ to find $w$.