a bookstore is placing an order for new books and magazines each book costs 15$ and each magazine cost 5$ the store can spend at most $1,000 on these books and magazines additionally each book takes up 2 square feet of space and each magazine takes up 1 square feet of space. the store has only 50 square feet of display space avaiable for the new items the following system of inequalities can be used to determine possible combinations of books x and magazines y the store can order

Question

what combination of books and magazines can the bookstore order

Answer 1

To find the combination of books \( x \) and magazines \( y \) that the bookstore can order, we can set up the following system of inequalities based on the constraints given:

1. Cost Constraint: Each book costs $15 and each magazine costs $5. The total spending cannot exceed $1,000: \[ 15x + 5y \leq 1000 \]

2. Space Constraint: Each book takes up 2 square feet and each magazine takes up 1 square feet. The total space cannot exceed 50 square feet: \[ 2x + y \leq 50 \]

3. Non-negativity Constraints: The number of books and magazines must be non-negative: \[ x \geq 0 \] \[ y \geq 0 \]

With these inequalities, we can now find the feasible region, which will describe the possible combinations of books and magazines.

Step 1: Graphing the Inequalities

1. Graph the first inequality:

   • Rearranging \( 15x + 5y \leq 1000 \) gives: \[ 5y \leq 1000 - 15x \quad \Rightarrow \quad y \leq 200 - 3x \]
   • Intercepts:
     • If \( x = 0 \): \( y = 200 \) (intercept on y-axis)
     • If \( y = 0 \): \( 15x = 1000 \Rightarrow x = \frac{1000}{15} \approx 66.67 \) (intercept on x-axis)

2. Graph the second inequality:

   • Rearranging \( 2x + y \leq 50 \) gives: \[ y \leq 50 - 2x \]
   • Intercepts:
     • If \( x = 0 \): \( y = 50 \) (intercept on y-axis)
     • If \( y = 0 \): \( 2x = 50 \Rightarrow x = 25 \) (intercept on x-axis)

Step 2: Identifying the Feasible Region

To determine the feasible region, graph both lines on a coordinate plane, and shade the region that satisfies both inequalities. The feasible region is the area where the shaded parts overlap and is bounded by the axes.

Step 3: Finding Corner Points

Calculate the corner points of the feasible region. These points can be found by solving the system of equations defined by the lines:

1. From \( y = 200 - 3x \) and \( y = 50 - 2x \): \[ 200 - 3x = 50 - 2x \quad \Rightarrow \quad x = 150 \] Plugging \( x = 150 \) back to find \( y \): \[ y = 200 - 3(150) = 200 - 450 = -250 \quad (\text{not feasible since } y<0) \]

Since only non-negative values are acceptable, we'll check the x and y intercepts we determined earlier and where they satisfy both inequalities.

From \( 15x + 5y = 1000 \):

• If \( x = 0 \), then \( y = 200 \) (but not valid due to space constraint).
• If \( y = 0 \), calculate \( x = 66.67 \) (but not valid due to space constraint).

From \( 2x + y = 50 \):

• If \( x = 0 \), \( y = 50 \) (valid).
• If \( y = 0 \), \( x = 25 \) (valid).

Step 4: Calculate Other Intersection Points

Systems resulting from inequalities need valid testing. Feasible Points of Intersection are:

• (0, 50)
• (25, 0)
• Where lines cross within the defined constraints.

Conclusion

The bookstore can order combinations of \( x \) books and \( y \) magazines from feasible points \( (0, 0), (0, 50), (25, 0) \).

All combinations that fit:

• \( x \) Books \( 0 \leq x \leq 25 \)
• \( y \) Magazines fitting remaining from the budget.

By evaluating corner points not exceeding the budget with \( x \cdot 15 + y \cdot 5 \leq 1000. \)

The final feasible combinations would require checking against the derived boundaries. Full exhaustive enumeration might be necessary within fine margin limits comparing display constraints.