A set of constraints and a profit function are given. Answer parts (a) and (b). \( x+y \leq 5 \) \( x+3 y \leq 9 \) \( x \geq 0 \) \( y \geq 0 \) \( P=5 x+4 y \) a) Graph the constraint s, shade the feasible region, and determine its vertices.

We start with the constraints:  1) x + y ≤ 5  2) x + 3y ≤ 9  3) x ≥ 0  4) y ≥ 0  Profit function: P = 5x + 4y Step 1. Graph the constraints.  • For x + y = 5, the line crosses the x-axis at (5, 0) and the y-axis at (0, 5). Since x + y ≤ 5, the area below (or on) this line is part of the feasible region.  • For x + 3y = 9, the line crosses the x-axis at (9, 0) and the y-axis at (0, 3). Since x + 3y ≤ 9, the area below (or on) this line is included.  • x ≥ 0 and y ≥ 0 mean that only the first quadrant is considered. Step 2. Determine the vertices of the feasible region.  A. Intersection with the axes:   – The origin (0, 0) is an obvious vertex.   – With x = 0:    • x + y = 5 gives (0, 5) but note that (0, 5) must also satisfy x + 3y ≤ 9. Check: 0 + 3(5) = 15 > 9, so (0, 5) is outside.    • x + 3y = 9 gives (0, 3). And (0, 3) satisfies x + y = 3 ≤ 5.  B. Intersection of x + y = 5 and y = 0 gives (5, 0), and it satisfies x + 3y = 5 ≤ 9.  C. Intersection of the two lines:   Solve x + y = 5 and x + 3y = 9 simultaneously.   Subtract the first equation from the second:    (x + 3y) – (x + y) = 9 – 5    2y = 4    y = 2   Substitute y = 2 in x + y = 5:    x = 5 – 2 = 3   This gives the vertex (3, 2).  D. Finally, we note that (9, 0) is the intercept of x + 3y = 9, but it does not satisfy x + y ≤ 5 because 9 > 5. Step 3. List the vertices of the feasible region.  The feasible region is the polygon with vertices:   • (0, 0)   • (0, 3)   • (3, 2)   • (5, 0) Step 4. Graphing:  On a coordinate system, draw the line x + y = 5 connecting (5, 0) and (0, 5) and only consider the part in the first quadrant. Similarly, draw the line x + 3y = 9 connecting (9, 0) and (0, 3). Then shade the area that satisfies both inequalities (the area below both lines) along with x ≥ 0 and y ≥ 0. The intersection of these shaded regions is your feasible region. Answer (a): • The feasible region is the shaded region in the first quadrant bounded by the lines x + y = 5 and x + 3y = 9. • Its vertices are (0, 0), (0, 3), (3, 2), and (5, 0).