Step by step
The question
Question: minimize C = 4x + 5y, when x ≥ 0, y ≥ 0, x + 2y ≥ 6, 2x + y ≥ 6.
Minimize means: make C as small 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 + 2y ≥ 6 is the line x + 2y = 6.
It crosses the axes at (6, 0) and (0, 3).
Solid: "or equal to".
The edge of 2x + y ≥ 6 is the line 2x + y = 6.
It crosses the axes at (3, 0) and (0, 6).
Solid: "or equal to".
2 Shade the region
x ≥ 0 and y ≥ 0: right of the y-axis, above the x-axis.
"Greater than or equal": the side away from (0, 0).
The shaded part fits every rule.
3 Find the corners
The corners are where two edges of the region meet.
Where x + 2y = 6 meets 2x + y = 6: we solve the two together, as in the systems lesson.
x = 2, y = 2.
4 Test each corner
At (0, 6): C = 30.
At (6, 0): C = 24.
At (2, 2): C = 18.
Careful!
The smallest value is always at a corner of the shaded region.
So we only test the corners.
5 Pick the best
The smallest is 18, at x = 2, y = 2.
6 Check
Let's check with a point inside the region, (3, 3): C is 27.
That is more than 18.
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.