# Linear Programming-Need help with one problem

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• November 16th 2007, 08:49 PM
soly_sol
Linear Programming-Need help with one problem

An objective function and a system of linear inequalities representing constraints are given. Graph the system of inequalities representing the constraints. Find the value of the objective function at each corner of the graphed region. Use these values to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.

Objective Function z = 19x + 4y
Constraints 0 < or equal to x < or equal to 10
0 < or equal to y < or equal to 5
3x + 2y > or equal to 6
• November 17th 2007, 04:44 AM
earboth
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Quote:

Originally Posted by soly_sol

An objective function and a system of linear inequalities representing constraints are given. Graph the system of inequalities representing the constraints. Find the value of the objective function at each corner of the graphed region. Use these values to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.

Objective Function z = 19x + 4y
Constraints 0 < or equal to x < or equal to 10
0 < or equal to y < or equal to 5
3x + 2y > or equal to 6

Hello,

the objective function has a straight line as it's graph and the value for z correspond with the y-intercept of this line:

$z = 19x + 4y~\implies~y=-\frac{19}{4}x+\underbrace{\frac z4}_{\text{y-intercept}}$ . That means: Take the y-intercept from the drawing and multiply it by 4 to get z.

Constraints:
$\left \{ \begin{array}{l}x\geq 0 \wedge x \leq 10 \\ y\geq0 \wedge y\leq 5 \\y \geq -\frac32 x + 3\end{array} \right.$ These inequaltities will give a pentagon.

Now draw parallel lines through the vertices of the pentagon which have the slope $m = -\frac{19}{4}$. The greater the y-intercept the greater the value for z. If you use the point (10, 5) then the function with the greatest y-intercept is:

$y=-\frac{19}{4}x+\frac {210}{4}$ . Because $\frac14 z = \frac {210}{4} ~\implies~z=210$

Remark:
(1) I forgot to draw the line: $y = -\frac{19}{4} x +5$
(2) the axes have different scales!
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