Question

Solve the following linear programming problem graphically:
Maximize $Z = 4x + y$ subjected to constraints:
$x + y \leqslant 50$
$3x + y \leqslant 90$
$x \geqslant 0$
$y \geqslant 0$

Answer 1

Hint: Draw the lines for the equations $x + y = 50$, $3x + y = 90$, $x = 0$, and $y = 0$. And shade the region for the given inequality and then find the common region. Find the corner points of the common region and evaluate the value of Z at each of these points and find the maximum value.

Complete step-by-step answer:
We need to maximize $Z = 4x + y$ subject to the given constraints as follows graphically:
$x + y \leqslant 50$
$3x + y \leqslant 90$
$x \geqslant 0$
$y \geqslant 0$
For the first equation, we have as follows:
$x + y = 50$
We plot the line with the following points:

x	y
0	50
50	0

Since, the origin satisfies the inequality, we shade towards the origin.
For the second equation, we have as follows:
$3x + y = 90$
We plot the line with the following points:

x	y
0	90
30	0

Since, the origin satisfies the inequality, we shade towards the origin.
For $x \geqslant 0$ and $y \geqslant 0$ it just represents the first quadrant, hence, we have the graph as follows:

The corner points are O, B, F and D. We evaluate Z at these points:

Points	Z = 4x + y
O (0, 0)	0
B (0, 50)	50
F (20, 30)	110
D (30, 0)	120

Hence, Z has a maximum value of 120 at the point (30, 0).

Note: You can cross-check the answer for the given point by substituting in the constraint equations and check if they satisfy the inequality, if they don’t then, there is some mistake in the graphing.