Question

ben makes only tables and chairs at his wood-making shop.

• ben is not able to make more than 500 pieces of furniture per month.
• ben needs to make more chairs than tables.
• ben needs to make at least 25 tables and 100 chairs each month.

let x represent the number of tables and y the number of chairs they make per month.
which system describes the monthly requirements of bens wood-making shop?
a. $\

$$\begin{cases}x + y \\leq 500 \\\\ y > x \\\\ x \\geq 25 \\\\ y \\geq 100\\end{cases}$$

$
c. $\

$$\begin{cases}x + y \\geq 500 \\\\ y > x \\\\ x \\leq 25 \\\\ y \\leq 100\\end{cases}$$

$
b. $\

$$\begin{cases}x + 4y \\leq 500 \\\\ y > x \\\\ x \\geq 25 \\\\ y \\geq 100\\end{cases}$$

$
d. $\

$$\begin{cases}x + 4y \\geq 500 \\\\ y > x \\\\ x \\leq 25 \\\\ y \\leq 100\\end{cases}$$

$

Explanation:

1. Analyze the total furniture constraint: Ben can't make more than 500 pieces, so \( x + y \leq 500 \) (since \( x \) is tables and \( y \) is chairs, total is \( x + y \)).
2. Analyze the chairs vs tables constraint: Ben needs more chairs than tables, so \( y > x \).
3. Analyze the minimum tables constraint: At least 25 tables, so \( x \geq 25 \).
4. Analyze the minimum chairs constraint: At least 100 chairs, so \( y \geq 100 \).

Now check the options:

• Option A: \( x + y \leq 500 \), \( y > x \), \( x \geq 25 \), \( y \geq 100 \) – matches all constraints.
• Option B: \( x + 4y \leq 500 \) is incorrect (no reason for 4y, total is \( x + y \)).
• Option C: \( x + y \geq 500 \) (wrong, should be ≤), \( x \leq 25 \) (wrong, should be ≥), \( y \leq 100 \) (wrong, should be ≥).
• Option D: \( x + 4y \geq 500 \) (wrong), \( x \leq 25 \) (wrong), \( y \leq 100 \) (wrong).

Answer:

A. \(

$$\begin{cases} x + y \leq 500 \\ y > x \\ x \geq 25 \\ y \geq 100 \end{cases}$$

\)