Maximize P = 2x + 5y: x ≥ 0, y ≥ 0, x + y ≤ 8, y ≤ 6 Step by Step

Step by step

The question

Question: maximize P = 2x + 5y, when x ≥ 0, y ≥ 0, x + y ≤ 8, y ≤ 6.

maximize P = 2x + 5y

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

x and y must follow every rule.

= x ≥ 0, y ≥ 0, x + y ≤ 8, y ≤ 6

1 Draw the lines

A grid for x and y.

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

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

Solid: "or equal to".

The edge of y ≤ 6: the line y = 6, flat.

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.

4 Test each corner

At (0, 0): P = 0.

(0, 0): P = 2(0) + 5(0) = 0

At (0, 6): P = 30.

(0, 6): P = 2(0) + 5(6) = 30

At (8, 0): P = 16.

(8, 0): P = 2(8) + 5(0) = 16

At (2, 6): P = 34.

(2, 6): P = 2(2) + 5(6) = 34

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 34, at x = 2, y = 6.

maximum P = 34 at (2, 6)

6 Check

Let's check with a point inside the region, (2, 3): P is 19.

That is less than 34.

So the corner is the best!

inside: (2, 3): P = 19 < 34

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