Solving a system of inequality

[brooce]

Homework Statement

<= means less than or equal to.

30 <= X <= 80
10 <= Y <= 30
X + Y <= 80



2. The attempt at a solution

I think I have an idea of how to do it, but I just want to be sure.
Can someone tell me if this is right?
For X, subtract 30 from both sides, leaving X<=50.
For Y, subtract 10 from both sides, leaving Y<=20.
So X(50)+Y(20)<=80
70<=80.

When I subtract from both sides, I just forget about that one inequality?


I also need to graph the Constraints and find the region of feasible solutions as well as all the corner points. I am also lost here. I think I got the graph down, but I'm lost on the corner points. I think I found 2 corner points, but that's it. (30,10) and (30,30).

If you guys could help me out a little here that would be great!

Thanks!

 

[symbolipoint]

Are you missing your function of variables X and Y ? Some f(x,y) ?

You almost describe one of those linear optimization problems in which when you graph the equation system, one of the corners will become the optimum solution. You then need to test each corner point to determine which point is this optimum.

 

[brooce]

Are you missing your function of variables X and Y ? Some f(x,y) ?

You almost describe one of those linear optimization problems in which when you graph the equation system, one of the corners will become the optimum solution. You then need to test each corner point to determine which point is this optimum.

Yes that is what the problem asks for in the end. But in order to find the optimal solution I need to find all the corner points, which is what I am having a problem with.

 

[symbolipoint]

Yes that is what the problem asks for in the end. But in order to find the optimal solution I need to find all the corner points, which is what I am having a problem with.

Draw the graph. The corner points will be clearly observable.

 

[brooce]

Not exactly. Not all the corner points will be a given. You have to solve the system of inequalities to find the remaining corner point(s). And that's where I'm stuck at.

 

[symbolipoint]

Not exactly. Not all the corner points will be a given. You have to solve the system of inequalities to find the remaining corner point(s). And that's where I'm stuck at.

As I said, draw the graph of the system and the corners will be plainly observable. The region of the inequality system is bounded by four lines and the region will be a trapezoidal shape. Read the points from the graph.

 

[brooce]

Ok, well I guess what I'm also asking is how would I solve the system of equations to find X and Y?

 

[symbolipoint]

Ok, well I guess what I'm also asking is how would I solve the system of equations to find X and Y?

Test each corner point in your function (which you did not give) to find which point optimizes the function. Substitute the coordinates into the function and evaluate the function at each set of corner coordinates. You then pick the point, (x, y) which gives you your largest value (or could optimization sometimes mean "smallest value"? Optimization means best, maybe this means largest... depends on context of the problem)

any way, draw graph; test each corner point in your function, choose the point which optimizes the function.

 

[brooce]

Ohh ok. I was confused as to the function part. But I think this should be it...and i'll test the coordinates in there to find the optimal solution. C=15X+8Y

Thanks for helping out.