# How to find the maximum value subject to constraints

I am currently enrolled in a college algebra course and am having difficulty finding the solution to the following problem since it is not covered in our textbook or in class. Any helpful hints or advice on how to start would be appreciated. Thanks!

Find the maximum value of $$z = 2x + 5y$$ subject to the constraints

$x \geq 5$

$y \geq 4$

$4x+3y \leq 56$

$x+y \leq 16$

• This is a linear programming problem ! Graphing these constraints will make this easier to solve ! – alkabary May 11 '15 at 3:37
• This is definitely in either your textbook or class. – Euler....IS_ALIVE May 11 '15 at 4:40
• Try converting the inequality constraints into equality constraints by using slack and surplus variables. – John Joy May 11 '15 at 13:27
• You can transform the variables $x,y$ into $u,v$: $u=x-5$ and $v=y-4$. Thus $u,v \geq 0$ – callculus May 12 '15 at 15:38

Graphing the constraints in blue and the iso-$z$ lines in red, we get:
the values of $z$ increasing for larger $x$ and $y$, as expected from the formula. We can find the maximum $z$ at $(5,11)$.