Show that the solution set of the following system of linear equations has no solution:

$2 x + y \geq 8, x + 2 y \geq 10, x \geq o, y \geq 0$

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Solution

Converting the given inequation,we obtain,2x+y=8,x+2y=10,x=0,y=0

Region represented by $2 x + y \geq 8$

The line 2x+y=8 meets the coordinate axes at (0.8) and (4,0).We find that (0,0) does not satisfy inequation $2 x + y \geq 8$ .So the portion not containing the origin is represented by the inequation $2 x + y \geq 8$ .

Region represented by $x + 2 y \geq 10$

The line x+2y=10meets the coordinate axes at (0,5) and (10,0).We find that satisfies containing (0,0) does not satisfy inequation $x + 2 y \geq 10$ .

Region represented by $x \geq 0$

Clearly, $x \geq 0$ represents the region lying on the right side of y-axis.

Region represented by $y \geq 0$

Clearly, $y \geq 0$ represents the region lying on the right side of x-axis.

From graph we can see that the solution set satisfying the given inequalities is an unbounded region.

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