Maximize Z = 2x + 3y

Subject to

\[x + y \geq 1\]

\[10x + y \geq 5\]

\[x + 10y \geq 1\]

\[ x, y \geq 0\]

#### Solution

First, we will convert the given inequations into equations, we obtain the following equations:*x** *+ *y* = 1, 10*x** *+*y* = 5,* x** *+ 10*y* = 1, *x* = 0 and* y* = 0

Region represented by *x** *+ *y* ≥ 1:

The line *x** *+ *y* = 1 meets the coordinate axes at *A*(1, 0) and *B*(0,1) respectively. By joining these points we obtain the line *x** *+ *y* = 1.

Clearly (0,0) does not satisfies the inequation* x** *+ *y* ≥ 1. So,the region in *xy* plane which does not contain the origin represents the solution set of the inequation* x** *+ *y* ≥ 1.

Region represented by 10*x** *+*y* ≥ 5:

The line 10*x** *+*y* = 5 meets the coordinate axes at \[C\left( \frac{1}{2}, 0 \right)\] and *D*(0, 5) respectively. By joining these points we obtain the line

10*x** *+*y* = 5.Clearly (0,0) does not satisfies the inequation 10*x** *+*y* ≥ 5. So,the region which does not contains the origin represents the solution set of the inequation 10*x*+*y* ≥ 5.

Region represented by *x** *+ 10*y* ≥ 1:

The line *x** *+ 10*y* = 1 meets the coordinate axes at \[A\left( 1, 0 \right)\] and \[F\left( 0, \frac{1}{10} \right)\] respectively. By joining these points we obtain the line*x** *+ 10*y** *= 1.Clearly (0,0) does not satisfies the inequation *x** *+ 10*y* ≥ 1. So,the region which does not contains the origin represents the solution set of the inequation *x** *+ 10*y* ≥ 1.

Region represented by *x* ≥ 0 and* y* ≥ 0:

Since, every point in the first quadrant satisfies these inequations. So, the first quadrant is the region represented by the inequations *x* ≥ 0, and *y *≥ 0.

The feasible region determined by the system of constraints *x** *+ *y* ≥ 1, 10*x** *+*y* ≥ 5,* x** *+ 10*y* ≥ 1, *x* ≥ 0, and *y* ≥ 0, are as follows.

The feasible region is unbounded.Therefore, the maximum value is infinity i.e. the solution is unbounded.**Disclaimer**:

The obtained answer is for the given question. Answer in the book is 2.It would be 2 if the question is to minimize Z instead of to maximize Z.