For the following LPs, write the associated LPs in standard
form:

(a) Minimize

5×1 + x2 + 4×3

subject to 5×1 − 7×2 − 4×3 ≤ 2

4×1 + 7×2 + 9×3 = 11

6×1 − 2×2 ≤ 6 x1, x2, x3 ≥ 0

(b) Minimize −3×1 + 2×2 − 4×3

subject to −4×1 − 5×2 + x3 = 11

6×1 − x2 − 2×3 = 8

x1, x2 ≥ 0, x3 ≤ 0

(Hint: substitute x3 = −x 0 3 )

(c) Minimize

6×1 − 9×2 + 7×3

subject to

5×1 − 7×2 − 4×3 ≥ 6

4×1 + 7×2 + 9×3 = 4

6×1 − 2×2 ≤ −10

x1, x2, x3 ≥ 0

(d) Maximize

−3×1 + 3×2 − 2×3

subject to

x1 + 2×2 − 2×3 ≤ 11

2×1 − x2 + x3 = −1

2×1 − 3×2 + 4×3 ≥ 8

x1, x2, x3 ≥ 0

(e) Maximize

−3×1 + x2 − 4×3

subject to

x1 + 2×2 − x3 ≤ 12

2×1 − x2 + x3 = −2

2×1 − 3×2 + 4×3 ≥ 6

x1, x3 ≥ 0, x2 ≤ 0