Maximize $Z=3x+4y$,subject to the constraints: $x+y \leq 1, x \geq 0, y \geq 0$.

(D) To maximize $Z=3x+4y$ subject to the constraints $x+y \leq 1, x \geq 0, y \geq 0$:
$1$. Plot the inequalities on the Cartesian plane. The region defined by $x+y \leq 1, x \geq 0, y \geq 0$ is a triangle with vertices $O(0,0)$,$A(1,0)$,and $B(0,1)$.
$2$. Evaluate the objective function $Z=3x+4y$ at each corner point of the feasible region:

Corner Point	Value of $Z=3x+4y$
$O(0,0)$	$3(0)+4(0) = 0$
$A(1,0)$	$3(1)+4(0) = 3$
$B(0,1)$	$3(0)+4(1) = 4$

$3$. Comparing the values,the maximum value of $Z$ is $4$,which occurs at the point $(0,1)$.