How Would I Solve This ?

Question

3y≤-x+9

Y+2x≤8 Maximum for P=4x+y

x≥0,y≥0

Answer 1

Hi Davianna, plot the two inequalities, 3y ≤ -x + 9 and y + 2x ≤ 8. Additional information, x ≥ 0 and y ≥ 0. 

 

What I tell students is to solve the linear equation by replacing the inequality sign by an equal sign. This will find the x- and y-intercepts of the inequality.

 

x- and y-intercepts for 3y ≤ -x + 9:

 

Change the inequality to an equal sign:

 

3y = -x + 9

 

When x = 0, y = 3, or the point (0, 3).

When y = 0, x = 9, or the point (9, 0).

 

x- and y-intercepts for y + 2x ≤ 8:

 

Change the inequality to an equal sign:

 

y + 2x = 8

 

When x = 0, y = 8, or the point (0, 8)

When y = 0, x = 4, or the point (4, 0)

 

There is an additional point of interest, when x = 3 and y = 2, or the point (3, 2). This is the point where the two inequalities cross.

 

Finding the maximum:

 

There are three points to check for the maximum of the equation, p = 4x + y. These points are determined by the constraints from the two inequalities and that x and y are both greater than or equal to zero.

 

The three points are:

 

The y-intercept point for the inequality: 3y ≤ -x + 9. The point is: (0, 3).

 

The x-intercept point for the inequality: y + 2x ≤ 8. The point is: (4, 0).

 

The third point is where the two inequalities cross. The point is: (3, 2).

 

Checking point (0, 3) for a maximum:

 

p = 4 * 0 + 3, p = 3.

 

Checking point (3, 2) for a maximum:

 

p = 4 * 3 + 2, p = 14.

 

Checking point (4, 0) for a maximum:

 

p = 4 * 4 + 0, p = 16.

 

This point is the maximum.

 

Solution:

 

The point (4, 0) is the maximum for the given inequalities and constraints.

 

Questions?