Step by step
The question
Question: maximize P = 3x + 5y, when x ≥ 0, y ≥ 0, x + y ≤ 8, x + 3y ≤ 12, x ≤ 7.
Maximize means: make P as big as it can be.
x and y must follow every rule.
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 x + 3y ≤ 12 is the line x + 3y = 12.
It crosses the axes at (12, 0) and (0, 4).
Solid: "or equal to".
The edge of x ≤ 7: the line x = 7, 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.
Where x + y = 8 meets x + 3y = 12: we solve the two together, as in the systems lesson.
x = 6, y = 2.
4 Test each corner
At (0, 0): P = 0.
At (0, 4): P = 20.
At (7, 0): P = 21.
At (6, 2): P = 28.
At (7, 1): P = 26.
Careful!
The biggest value is always at a corner of the shaded region.
So we only test the corners.
5 Pick the best
The biggest is 28, at x = 6, y = 2.
6 Check
Let's check with a point inside the region, (4, 1): P is 17.
That is less than 28.
So the corner is the best!
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.