Maximize P = x + 3y: x ≥ 0, y ≥ 0, x + 2y ≤ 8, x ≤ 6 Step by Step

Step by step

The question

Question: maximize P = x + 3y, when x ≥ 0, y ≥ 0, x + 2y ≤ 8, x ≤ 6.

maximize P = x + 3y

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

x and y must follow every rule.

= x ≥ 0, y ≥ 0, x + 2y ≤ 8, x ≤ 6

1 Draw the lines

A grid for x and y.

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".

The edge of x ≤ 6: the line x = 6, straight up.

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 = 1(0) + 3(0) = 0

At (0, 4): P = 12.

(0, 4): P = 1(0) + 3(4) = 12

At (6, 0): P = 6.

(6, 0): P = 1(6) + 3(0) = 6

At (6, 1): P = 9.

(6, 1): P = 1(6) + 3(1) = 9

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 12, at x = 0, y = 4.

maximum P = 12 at (0, 4)

6 Check

Let's check with a point inside the region, (3, 1): P is 6.

That is less than 12.

So the corner is the best!

inside: (3, 1): P = 6 < 12

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