Question

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

$$\begin{cases}x \\geq 0\\\\x \\leq 4\\\\y \\geq -1\\\\y \\leq 5\\end{cases}$$

\\\\(\boldsymbol{p = 17}\\)\\(\boldsymbol{p = 3}\\)\\(\boldsymbol{p = 5}\\)\\(\boldsymbol{p = -1}\\)

Explanation:

Step1: Analyze the objective function

The objective function is \( P = 3x + y \). To maximize \( P \), we need to consider the maximum possible values of \( x \) and \( y \) within the constraints. Since the coefficient of \( x \) (3) is positive, increasing \( x \) will increase \( P \), and similarly, increasing \( y \) (with positive coefficient 1) will also increase \( P \).

Step2: Determine the maximum values of \( x \) and \( y \) from constraints

• For \( x \): The constraints are \( x \geq 0 \) and \( x \leq 4 \), so the maximum value of \( x \) is 4.
• For \( y \): The constraints are \( y \geq -1 \) and \( y \leq 5 \), so the maximum value of \( y \) is 5.

Step3: Substitute maximum \( x \) and \( y \) into \( P \)

Substitute \( x = 4 \) and \( y = 5 \) into \( P = 3x + y \):

$$ P = 3(4) + 5 = 12 + 5 = 17 $$

Answer:

\( P = 17 \)