# Linear Optimization: What is basic variables of the point where the solution is? For example: min: 8x+4y st: 3x+4y >= 6 5x + 2y <= 10 x+4y <= 4 x >= 0 y >= 0 After finding the...

Linear Optimization: What is basic variables of the point where the solution is?

For example:

min: 8x+4y

st:

3x+4y >= 6

5x + 2y <= 10

x+4y <= 4

x >= 0

y >= 0

After finding the solution : x=2; y=0; z=16

What are the basic variables?

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### 1 Answer

The basic variables here refer to x and y. The variable that needs to be optimized (in this case, z) depends on the basic variables (x and y).

The variable that needs to be minimized is z = 8x + 4y. For a particular pair of values for x and y, z will obtain its minimal value.

This pair is a pair of coordinates for some point on the xy coordinate plane.

This point has to be at one of the vertices of the region bounded by the graphs of the constraint equations:

The region bounded by the constraints is the smallest triangle in the first quadrant, bounded by the three slanted lines. One of its vertices is (2,0). For these values of x and y (x = 2, y = 0), the variable z = 8x + 4y will be minimum. The minimum value of z is z = 8*2 + 4*0 = 16.

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So when the question asks what are the basic variables for that example problem, what should I put down?

Would it be x = 2; y=0?