Step by step
The question
Question: minimize C = 5x + 2y, when x ≥ 0, y ≥ 0, 3x + y ≥ 9, x + y ≥ 5.
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 3x + y ≥ 9 is the line 3x + y = 9.
It crosses the axes at (3, 0) and (0, 9).
Solid: "or equal to".
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".
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 3x + y = 9 meets x + y = 5: we solve the two together, as in the systems lesson.
x = 2, y = 3.
4 Test each corner
At (0, 9): C = 18.
At (5, 0): C = 25.
At (2, 3): C = 16.
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 16, at x = 2, y = 3.
6 Check
Let's check with a point inside the region, (3, 4): C is 23.
That is more than 16.
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.