# A linear programming problem

Please guide me on the following question.

Consider the LP problem

maximize $x_1+x_2$

subject to

$x_1-2x_2\le10$

$x_2-2x_1\le10$

$x_1,x_2\ge0$

Which of the following is true?

$1.$ The LP problem admits an optimal solution.

$2.$ The LP problem is unbounded.

$3.$ The LP problem admits no feasible solution.

$4.$ The LP problem admits a unique feasible solution.

The first line passes through $(0,-5)$ and $(10,0)$. The second line passes through $(0,10)$ and $(-5,0)$. They both intersect at (-10,-10). Thereby, I am getting that it would be an unbounded problem and won't have any feasible solution.

That is, according to me, $2nd$ and $3rd$ options are correct. But answer should be only one option. Please help.

• It may be good for you to plot the feasible region - i.e. shade it. For e.g. check $(5, 5)$ - why would you think there is no feasible region? – Macavity Aug 16 '13 at 15:28
• got it. I earlier thought only vertices can give us a feasible solution. But ur comment makes me think that a region can be feasible solution. that makes sense. so answer is 2. thanks! – Ramit Aug 16 '13 at 16:00
• @Ramit: The intersection of the all areas left after the constraints give the feasible solutions. – Nick Jan 30 '15 at 15:14

Put $x_1 = x_2 \geq 0$ and check if this is a feasible solution. If so, then what happens to the objective function as $x_1 = x_2$ becomes larger and larger?