Find the maximum value of \[ P=9 x+8 y \] subject to the following constraints: Now find the profit, P, at each corner point using the above function. \[ \begin{array}{lll} x & y & P \\ 0 & 0 & {[?]} \\ 6 & 0 & \\ 0 & 7 \\ 3 & 4 \end{array} \quad\left\{\begin{array}{l} 8 x+6 y \leq 48 \\ 7 x+7 y \leq 49 \\ x \geq 0 \\ y \geq 0 \end{array}\right. \]

Step 1 ：Substitute \(x = 0\) and \(y = 0\) into the function \(P = 9x + 8y\). We get \(P = 9(0) + 8(0) = 0\).

Step 2 ：Substitute \(x = 6\) and \(y = 0\) into the function \(P = 9x + 8y\). We get \(P = 9(6) + 8(0) = 54\).

Step 3 ：Substitute \(x = 0\) and \(y = 7\) into the function \(P = 9x + 8y\). We get \(P = 9(0) + 8(7) = 56\).

Step 4 ：Substitute \(x = 3\) and \(y = 4\) into the function \(P = 9x + 8y\). We get \(P = 9(3) + 8(4) = 59\).

Step 5 ：The values of \(P\) at the corner points are: \[\begin{array}{lll} x & y & P \\ 0 & 0 & 0 \\ 6 & 0 & 54 \\ 0 & 7 & 56 \\ 3 & 4 & 59 \end{array}\]

Step 6 ：The maximum value of \(P\) under the given constraints is \(\boxed{59}\) at the point \((3,4)\).