Maximize P = 3x + 4y: x ≥ 0, y ≥ 0, 2x + y ≤ 10, x + 2y ≤ 8 Step by Step

Step by step

The question

Question: maximize P = 3x + 4y, when x ≥ 0, y ≥ 0, 2x + y ≤ 10, x + 2y ≤ 8.

maximize P = 3x + 4y

Maximize means: make P as big as it can be.

x and y must follow every rule.

= x ≥ 0, y ≥ 0, 2x + y ≤ 10, x + 2y ≤ 8

1 Draw the lines

A grid for x and y.

The edge of 2x + y ≤ 10 is the line 2x + y = 10.

It crosses the axes at (5, 0) and (0, 10).

Solid: "or equal to".

The edge of x + 2y ≤ 8 is the line x + 2y = 8.

It crosses the axes at (8, 0) and (0, 4).

Solid: "or equal to".

2 Shade the region

x ≥ 0 and y ≥ 0: right of the y-axis, above the x-axis.

"Less than or equal": the side of each line with (0, 0).

The shaded part fits every rule.

3 Find the corners

The corners are where two edges of the region meet.

Where 2x + y = 10 meets x + 2y = 8: we solve the two together, as in the systems lesson.

x = 4, y = 2.

2x + y = 10 and x + 2y = 8: x = 4, y = 2

4 Test each corner

At (0, 0): P = 0.

(0, 0): P = 3(0) + 4(0) = 0

At (0, 4): P = 16.

(0, 4): P = 3(0) + 4(4) = 16

At (5, 0): P = 15.

(5, 0): P = 3(5) + 4(0) = 15

At (4, 2): P = 20.

(4, 2): P = 3(4) + 4(2) = 20

Careful!

The biggest value is always at a corner of the shaded region.

So we only test the corners.

Careful: the best point is always a corner of the region

5 Pick the best

The biggest is 20, at x = 4, y = 2.

maximum P = 20 at (4, 2)

6 Check

Let's check with a point inside the region, (2, 1): P is 10.

That is less than 20.

So the corner is the best!

inside: (2, 1): P = 10 < 20

Let's look back.

First: draw the lines.

Then: shade the region.

Then: find the corners.

Then: test each corner.

Then: pick the best.

Last: check.

Answer