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(C) Step $1$: Identify the corner points of the feasible region defined by the constraints.Step $2$: The lines are $2x + 3y = 6$ and $x + y = 1$.Step

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$8$

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(C) Step $1$: Identify the corner points of the feasible region defined by the constraints.Step $2$: The lines are $2x + 3y = 6$ and $x + y = 1$.Step $3$: Intersection of $2x + 3y = 6$ and $x + y = 1$ gives $x = 3, y = -2$ (outside the first quadrant).Step $4$: The vertices of the feasible region are $(0, 1)$, $(0, 2)$, $(1, 0)$, and $(3, 0)$.Step $5$: Evaluate $z = 3x + y$ at each vertex:At $(0, 1)$, $z = 3(0) + 1 = 1$.At $(0, 2)$, $z = 3(0) + 2 = 2$.At $(1, 0)$, $z = 3(1) + 0 = 3$.At $(3, 0)$, $z = 3(3) + 0 = 9$.Step $6$: Maximum value is $9$ and minimum value is $1$.Step $7$: The difference is $9 - 1 = 8$.

The difference between the maximum value and the minimum value of the objective function $z = 3x + y$ subject to the constraints $2x + 3y \leq 6$, $x + y \geq 1$, $x \geq 0$, $y \geq 0$ is....

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