Question

1. maximize the equation $p = 4x + 3y$ using the constraints shown.\\\

$$\begin{cases}y \\geq 1.5x - 3 \\\\x + y \\leq 7 \\\\x \\geq 0 \\\\y \\geq 0\\end{cases}$$

\\\\(\boldsymbol{p = 25}\\)\\(\boldsymbol{p = 24}\\)\\(\boldsymbol{p = 21}\\)\\(\boldsymbol{p = 0}\\)

Explanation:

Step1: Find Intersection Points

First, find the intersection of \( y = 1.5x - 3 \) and \( x + y = 7 \). Substitute \( y = 7 - x \) into \( y = 1.5x - 3 \):
\( 7 - x = 1.5x - 3 \)
\( 7 + 3 = 1.5x + x \)
\( 10 = 2.5x \)
\( x = 4 \), then \( y = 7 - 4 = 3 \). So one vertex is \( (4, 3) \).

Next, find intersection of \( x + y = 7 \) with \( x = 0 \): \( y = 7 \), vertex \( (0, 7) \).

Intersection of \( x + y = 7 \) with \( y = 0 \): \( x = 7 \), but check \( y \geq 1.5x - 3 \): \( 0 \geq 1.5(7) - 3 = 7.5 \), false. So find intersection of \( y = 1.5x - 3 \) with \( y = 0 \): \( 0 = 1.5x - 3 \Rightarrow x = 2 \), vertex \( (2, 0) \).

Intersection of \( x = 0 \) and \( y = 0 \): \( (0, 0) \), but check \( y \geq 1.5x - 3 \): \( 0 \geq -3 \), true. But also check \( x + y \leq 7 \), true.

Step2: Evaluate \( P = 4x + 3y \) at Vertices

• At \( (4, 3) \): \( P = 4(4) + 3(3) = 16 + 9 = 25 \)
• At \( (0, 7) \): \( P = 4(0) + 3(7) = 21 \)
• At \( (2, 0) \): \( P = 4(2) + 3(0) = 8 \)
• At \( (0, 0) \): \( P = 0 \)

Answer:

\( P = 25 \) (corresponding to the option \( P = 25 \))