Worked examples
The question
Question: maximize P = 3x + 2y, when x ≥ 0, y ≥ 0, x + y ≤ 6, x ≤ 4.
maximize P = 3x + 2y
Maximize means: make P as big as it can be.
x and y must follow every rule.
= x ≥ 0, y ≥ 0, x + y ≤ 6, x ≤ 4
1 Draw the lines
A grid for x and y.
The edge of x + y ≤ 6 is the line x + y = 6.
It crosses the axes at (6, 0) and (0, 6).
Solid: "or equal to".
The edge of x ≤ 4: the line x = 4, 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.
4 Test each corner
At (0, 0): P = 0.
(0, 0): P = 3(0) + 2(0) = 0
At (0, 6): P = 12.
(0, 6): P = 3(0) + 2(6) = 12
At (4, 0): P = 12.
(4, 0): P = 3(4) + 2(0) = 12
At (4, 2): P = 16.
(4, 2): P = 3(4) + 2(2) = 16
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 16, at x = 4, y = 2.
maximum P = 16 at (4, 2)
6 Check
Let's check with a point inside the region, (2, 2): P is 10.
That is less than 16.
So the corner is the best!
inside: (2, 2): P = 10 < 16
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 maximum P = 16 at (4, 2)
▶ Watch it solved
The question
Question: maximize P = 2x + 3y, when x ≥ 0, y ≥ 0, x + y ≤ 6, 2x + y ≤ 8.
maximize P = 2x + 3y
Maximize means: make P as big as it can be.
x and y must follow every rule.
= x ≥ 0, y ≥ 0, x + y ≤ 6, 2x + y ≤ 8
1 Draw the lines
A grid for x and y.
The edge of x + y ≤ 6 is the line x + y = 6.
It crosses the axes at (6, 0) and (0, 6).
Solid: "or equal to".
The edge of 2x + y ≤ 8 is the line 2x + y = 8.
It crosses the axes at (4, 0) and (0, 8).
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 = 6 meets 2x + y = 8: we solve the two together, as in the systems lesson.
x = 2, y = 4.
x + y = 6 and 2x + y = 8: x = 2, y = 4
4 Test each corner
At (0, 0): P = 0.
(0, 0): P = 2(0) + 3(0) = 0
At (0, 6): P = 18.
(0, 6): P = 2(0) + 3(6) = 18
At (4, 0): P = 8.
(4, 0): P = 2(4) + 3(0) = 8
At (2, 4): P = 16.
(2, 4): P = 2(2) + 3(4) = 16
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 18, at x = 0, y = 6.
maximum P = 18 at (0, 6)
6 Check
Let's check with a point inside the region, (1, 2): P is 8.
That is less than 18.
So the corner is the best!
inside: (1, 2): P = 8 < 18
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 maximum P = 18 at (0, 6)
▶ Watch it solved
The question
Question: minimize C = 2x + 3y, when x ≥ 0, y ≥ 0, x + y ≥ 4, x + 3y ≥ 6.
minimize C = 2x + 3y
Minimize means: make C as small as it can be.
x and y must follow every rule.
= x ≥ 0, y ≥ 0, x + y ≥ 4, x + 3y ≥ 6
1 Draw the lines
A grid for x and y.
The edge of x + y ≥ 4 is the line x + y = 4.
It crosses the axes at (4, 0) and (0, 4).
Solid: "or equal to".
The edge of x + 3y ≥ 6 is the line x + 3y = 6.
It crosses the axes at (6, 0) and (0, 2).
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 + y = 4 meets x + 3y = 6: we solve the two together, as in the systems lesson.
x = 3, y = 1.
x + y = 4 and x + 3y = 6: x = 3, y = 1
4 Test each corner
At (0, 4): C = 12.
(0, 4): C = 2(0) + 3(4) = 12
At (6, 0): C = 12.
(6, 0): C = 2(6) + 3(0) = 12
At (3, 1): C = 9.
(3, 1): C = 2(3) + 3(1) = 9
Careful!
The smallest 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 smallest is 9, at x = 3, y = 1.
minimum C = 9 at (3, 1)
6 Check
Let's check with a point inside the region, (4, 2): C is 14.
That is more than 9.
So the corner is the best!
inside: (4, 2): C = 14 > 9
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 minimum C = 9 at (3, 1)
▶ Watch it solved
