What does it means?

Question

This is a linear programming problem:
maximize z = 3x + y subject to
2x - y <= 4
2x + 3y <= 12
y <= 3
here,What does"z = 3x + y" represents?
please inform me.

I also want to know what's the importance of it in linear programming?

Answer 1

you have three equations and you want to maximize Z in first equation. based on value of z in your target equation (z=3x+y) can be a set of parallel line which should touch the up right side of the feasible area (an area which is defied by last three equations). The best way to solve this problem is to draw three lines and find the area for your answer and then draw your target line (z=3x+y) by guessing any value for z and move up the line parallel to the line that you drew to touch the upright of the area the intersection of the target line and the feasible area is the answer.
You may want to find your problem's answer here:
http://www.geocities.com/h_sadeghy/01.bmp

Answer 2

Apparently you have a three-dimensional programming problem. Here is how to solve it:
graph on the x,y-axes the subject to equations. Within that triad of equations is a zone that you should imagine goes down to minus infinity and up to positive infinity.
Now in 3-space graph z= 3x + y. Where the z cuts the 2-D equation triad is the area or zone of your solution set.

Answer 3

It's a multivariate function. If z = 3x + y, then x and y are independent variables while z is the dependent variable. You want z to have the largest possible variable, based on the restrictions placed on x and y.