Question

The maximum value of Z = 4x + 2y subjected to the constraints 2x + 3y ≤ 18, x + y ≥ 10; x, y ≥ 0 is

A. 36

B. 40

C. 20

D. None of these

Answer 1

Correct answer is D.

Given,

Z = 4x + 2y

Subjected to constraints,

2x + 3y ≤ 18

x + y ≥ 10

x ≥ 0

y ≥ 0

Consider, the inequalities as equalities for some time,

2x + 3y = 18 and x + y = 10,

If we convert these into intercept line format equations, we get,

[Dividing the whole equation with the right hand side number of the equation]

From this form of line, we can say that the line 2x + 3y = 18 meets the x-axis at (9, 0) and y-axis at (0, 6).

This shows the inequality 2x + 3y ≤ 18 holds good in the below blue colored region.

Similarly from the intercept line format, we can say that the line x + y = 10 meets the x-axis at (10, 0) and yaxis at (0, 10).

This shows the inequality x + y ≥ 10 holds above in the green colored region.

Now considering the inequalities, x ≥ 0 and y ≥ 0, this clearly shows the region where both x and y are positive. This represents the 1^st quadrant of the graph.

So, the solutions of the LPP are in the first quadrant where the inequalities meet.

Now by plotting both graphs 2x + 3y ≤ 18 and x + y ≥ 10 we get the below graph.

We can clearly see that, there is no area in the 1^st quadrant where the two inequalities met.

This clearly says that there is no solution for the LPP with the given constraints.