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Explanation & Answer
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1.
a. −3𝑥 + 𝑦 = −6, 𝑦 = 3𝑥 − 6, for 𝑥 = 1 it is 3 ∙ 1 − 6 = −𝟑.
3
3
b. 2𝑥 + 3𝑦 = 12, 2𝑥 = 12 − 3𝑦, 𝑥 = 6 − 2 𝑦, for 𝑦 = 2 it is 6 − 2 ∙ 2 = 6 − 3 = 𝟑.
2. They are inequalities, not equations.
a. {
𝑥+𝑦 ≥2
. The graph is below (the solutions are in the darkest area).
2𝑥 − 𝑦 < 4
𝑥 − 2𝑦 ≤ 6
𝑥 − 2𝑦 ≤ 6
, which is the same as {
. The graph is below (the solutions are
2𝑥 − 4𝑦 ≥ 0
𝑥 − 2𝑦 ≥ 0
in the darkest area).
b. {
3.
𝑥≥0
𝑦≥0
a. Maximize 𝑃 = 2𝑥 + 𝑦 subject to {
.
𝑥+𝑦 ≤6
𝑥+𝑦 ≥1
The graph is below. We see that the maximum is reached at the angle (𝟔, 𝟎) (the orange
line is 2𝑥 + 𝑦 = 15, the red is 2𝑥 + 𝑦 = 12 — the level line that touches the feasible region).
The maximum value is 𝑃(6,0) = 2 ∙ 6 + 0 = 𝟏𝟐.
𝑥≥0
𝑦≥0
𝑥≤6 .
b. Minimize 𝐶 = 2𝑥 + 3 subject to
𝑦≤5
{𝑥 + 𝑦 ≥ 2
The graph is below. We see that the minimum is reached at the entire interv...