Answer: 20
Step-by-step explanation:
- Step 1: Find the vertices (intersections) of the constraints by either graphing or solving the system of equations. I graphed them and found two of the vertices: (0, 1) and (4,0) but needed to solve the system to find the third vertex (point of intersection).
-x + 3 = [tex]\frac{1}{3}x[/tex] + 1
+x -1 +x -1
2 = [tex]\frac{4}{3}x[/tex]
(3)2 = [tex](3)\frac{4}{3}x[/tex]
6 = 4x
÷4 ÷4
[tex]\frac{3}{2}[/tex] = x
y = [tex]\frac{1}{3}x[/tex] + 1
= [tex]\frac{1}{3}(\frac{3}{2)}[/tex] + 1
= [tex]\frac{1}{2}[/tex] + 1
= [tex]\frac{3}{2}x[/tex]
Third vertex is at [tex](\frac{3}{2},\frac{3}{2)}[/tex]
- Step 2: Input the vertices into the objective function: C = 5x - 4y
(0, 1): C = 5(0) - 4(1)
= 0 - 4
= -4
[tex](\frac{3}{2},\frac{3}{2)}[/tex]: C = 5[tex](\frac{3}{2})[/tex] - 4[tex](\frac{3}{2})[/tex]
= [tex]\frac{15}{2}[/tex] - [tex]\frac{12}{2}[/tex]
= [tex]\frac{3}{2}[/tex]
= 1.5
(4, 0): C = 5(4) - 4(0)
= 20 - 0
= 20
- Step 3: Evaluate the answers above to find the maximum: 20
