# What's a basic solution, and how do we find them?

I've just started learning linear programming, and for some reason, have run into a question about something that isn't mentioned in the first chapter (and we're supposed to answer these questions based on the first chapter).

What is a "basic" solution? It is only mentioned once (so far) in the book, in the following manner:

$$\text{"The solutions we obtain by setting the nonbasic variables to zero are called basic feasible solutions"}$$

But the question now asks; "indicate EACH basic solution, and determine which are feasible and which are infeasible", and I don't see how the above quote says anything about several basic solutions. Surely there's only one way we can set all variables to zero, and so only one basic solution?

For example, what are the basic solutions for something as simple as; $$max \ 2x + y \\ s.t. x + y \le 3 \\ x \le 2$$ where both variables are nonnegative.

• You should read definitions. Perhaps "basic solution" is one on the vertices of the domain, which in this case are $\;(0,0)\,,\,\,(2,0)\,,\,\,(2,1)\,,\,\,(0,3)\;$ , so you probably have to evaluate the target function $\;f(x,y)=2x+y\;$ on each of these points and choose the max/min value. – Timbuc Feb 2 '15 at 13:43
• BTW, what book are you reading on this? – Timbuc Feb 2 '15 at 13:50
• Vanderbel, Linear Programming. It's also not using matrices and linear algebra (but again, have only read the first two chapters), so I can't get much help from reading online, where apparently "basic solution" has something to do with linearly independent columns in a matrix. – Arda Daniels Feb 2 '15 at 13:53

Rewrite the inequalities as $$x+y+a=3\\x+b=2\\x\geq0,y\geq0,a\geq0,b\geq0$$
A basic solution has any two of $a,b,x,y$ equal to zero. The other two variables are forced by the two equations.
• So set x and b equal to zero. The second equality gives $0 + 0= 2$....So what does that mean? – Arda Daniels Feb 2 '15 at 13:37