Question

\[{\text{Maximise }}z{\text{ }} = {\text{ }}x{\text{ }} + {\text{ }}y,{\text{ subject to }}x{\text{ }} - {\text{ }}y{\text{ }} \leqslant {\text{ }} - 1,{\text{ }} - x{\text{ }} + {\text{ }}y{\text{ }} \leqslant {\text{ }}0,{\text{ }}x{\text{ }},{\text{ }}y{\text{ }} \geqslant {\text{ }}0\]

Answer 1

Hint: We can see all the points are present in a X-Y plane. We will plot, these points and mark the area:
For ‘≥’ sign area above the line whereas for ‘≤’ sign area below the line is considered.
To maximise the value, we need to consider the area lying above the same.

Complete step-by-step answer:
According to the question, we have to maximize \[z{\text{ }} = {\text{ }}x{\text{ }} + {\text{ }}y\]
Subjected to:
 \[x{\text{ }} - {\text{ }}y{\text{ }} \leqslant {\text{ }} - 1\]
\[ - x{\text{ }} + {\text{ }}y{\text{ }} \leqslant {\text{ }}0\]
\[{\text{ }}x{\text{ }},{\text{ }}y{\text{ }} \geqslant {\text{ }}0\]
For finding the areas to be considered on the graph, we need to plot some points on the graph and these points can be calculated by equating either variable to 0:
For \[x{\text{ }} - {\text{ }}y{\text{ }} \leqslant {\text{ }} - 1:\]:

x	0	-1
y	1	0

For \[ - x{\text{ + }}y{\text{ }} \leqslant {\text{ 0:}}\]:

x	0	1
y	0	1

Plotting these values on graph:

The arrows show the area that is to be considered by each line.
We can observe that there is no common or feasible region between the given lines.
This implies that there is no maximum value of Z.

Note: Feasible region is that region where all the points of subjects and constraints are satisfied.Be careful while taking the area into consideration based upon the signs of inequality present.Instead of colouring and shading the regions, draw arrows (as drawn in figure) for better clarification.