Maximize P = x + 3y: x ≥ 0, y ≥ 0, x + y ≤ 5, y ≤ 3 Step by Step

Step by step

The question

Question: maximize P = x + 3y, when x ≥ 0, y ≥ 0, x + y ≤ 5, y ≤ 3.

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 + y ≤ 5, y ≤ 3

1 Draw the lines

A grid for x and y.

The edge of x + y ≤ 5 is the line x + y = 5.

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

Solid: "or equal to".

The edge of y ≤ 3: the line y = 3, 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 = 1(0) + 3(0) = 0

At (0, 3): P = 9.

(0, 3): P = 1(0) + 3(3) = 9

At (5, 0): P = 5.

(5, 0): P = 1(5) + 3(0) = 5

At (2, 3): P = 11.

(2, 3): P = 1(2) + 3(3) = 11

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 11, at x = 2, y = 3.

maximum P = 11 at (2, 3)

6 Check

Let's check with a point inside the region, (1, 1): P is 4.

That is less than 11.

So the corner is the best!

inside: (1, 1): P = 4 < 11

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