Question

find the maximum value of
p = 9x + 8y
subject to the following constraints:
now, find the coordinates of the corner points.
(there is a table with x and y columns, some filled values, and a system of inequalities: 8x + 6y ≤ 48, 7x + 7y ≤ 49, x ≥ 0, y ≥ 0, and a graph with a purple region and two lines.)

Explanation:

Step1: Identify the intersection of the two main constraints

We have the inequalities \(8x + 6y \leq 48\) and \(7x + 7y \leq 49\). To find the intersection point (the corner point we need to calculate), we solve the system of equations formed by the equalities \(8x + 6y = 48\) and \(7x + 7y = 49\). First, simplify the second equation: divide both sides of \(7x + 7y = 49\) by 7, we get \(x + y = 7\), so \(y = 7 - x\).

Step2: Substitute \(y = 7 - x\) into the first equation

Substitute \(y = 7 - x\) into \(8x + 6y = 48\):
\[

$$\begin{align*} 8x + 6(7 - x)&= 48\\ 8x + 42 - 6x&= 48\\ 2x + 42&= 48\\ 2x&= 48 - 42\\ 2x&= 6\\ x&= 3 \end{align*}$$

\]

Step3: Find the value of \(y\)

Now that we have \(x = 3\), substitute \(x = 3\) into \(y = 7 - x\):
\(y = 7 - 3 = 4\)

Answer:

The \(x\)-coordinate of the corner point (the green box) is \(3\), and the \(y\)-coordinate is \(4\). So the corner point is \((3, 4)\).

To find the maximum value of \(P = 9x + 8y\), we evaluate \(P\) at each corner point:

• At \((0, 0)\): \(P = 9(0) + 8(0) = 0\)
• At \((6, 0)\): \(P = 9(6) + 8(0) = 54\)
• At \((0, 7)\): \(P = 9(0) + 8(7) = 56\)
• At \((3, 4)\): \(P = 9(3) + 8(4) = 27 + 32 = 59\)

The maximum value of \(P\) is \(59\).