For the following shaded region, the linear constraints are

Let's analyze the shaded region and its linear constraints:
1.
Looking at the graph, we can see:
- The region is bounded by the line \(y=x\)
- The line \(-x+3 y=3\)
- The \(x\)-axis \((y=0)\)
- The \(y\)-axis \((x=0)\)
2.
The shaded region shows:
- It's above the line \(y=x(\) so \(x-y \leq 0)\)
- It's below the line \(-x+3 y=3\)
- It's in the first quadrant \((x \geq 0, y \geq 0)\)
3.
Looking at the options given:
(A) \(x-y \leq 0,-x+3 y \leq 3, x \geq 0, y \geq 0\)
(B) \(x-y \geq 0,-x+3 y \geq 3, x \geq 0, y \geq 0\)
(C) \(x-y \geq 0,-x+3 y \leq 3, x \geq 0, y \geq 0\)
(D) \(x-y \leq 0,-x+3 y=3, x \geq 0, y \geq 0\)
The correct answer is \((A)\) because:
The region is above \(y=x\), so \(x-y \leq 0\)
The region is below \(-x+3 y=3\), so \(-x+3 y \leq 3\)
The region is in the first quadrant, so \(x \geq 0\) and \(y \geq 0\)
Therefore, the linear constraints defining the shaded region are:
\(x-y \leq 0,-x+3 y \leq 3, x \geq 0, y \geq 0\)
The shaded region is defined by the linear constraints from option (A):
\(\begin{aligned}
& x-y \leq 0 \\
& -x+3 y \leq 3 \\
& x \geq 0 \\
& y \geq 0
\end{aligned}\)