Maximize P = 2x + y: x ≥ 0, y ≥ 0, x + y ≤ 5, x + 3y ≤ 9 Step by Step

Step by step

The question

Question: maximize P = 2x + y, when x ≥ 0, y ≥ 0, x + y ≤ 5, x + 3y ≤ 9.

maximize P = 2x + y

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

x and y must follow every rule.

= x ≥ 0, y ≥ 0, x + y ≤ 5, x + 3y ≤ 9

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 x + 3y ≤ 9 is the line x + 3y = 9.

It crosses the axes at (9, 0) and (0, 3).

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 x + y = 5 meets x + 3y = 9: we solve the two together, as in the systems lesson.

x = 3, y = 2.

x + y = 5 and x + 3y = 9: x = 3, y = 2

4 Test each corner

At (0, 0): P = 0.

(0, 0): P = 2(0) + 1(0) = 0

At (0, 3): P = 3.

(0, 3): P = 2(0) + 1(3) = 3

At (5, 0): P = 10.

(5, 0): P = 2(5) + 1(0) = 10

At (3, 2): P = 8.

(3, 2): P = 2(3) + 1(2) = 8

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 10, at x = 5, y = 0.

maximum P = 10 at (5, 0)

6 Check

Let's check with a point inside the region, (2, 1): P is 5.

That is less than 10.

So the corner is the best!

inside: (2, 1): P = 5 < 10

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