Question

2.3 Solve the following linear programming graphically [5]

Minimize 𝑧 = 3𝑥 + 9𝑦

Subject to the constraints: 𝑥 + 3𝑦 ≥ 6

𝑥 + 𝑦 ≤ 10

𝑥 ≤ 𝑦

𝑥 ≥ 0; 𝑦 ≥ 0

Answer 1

Minimize $𝑧 = 3𝑥 + 9𝑦$

subject to the constraints

$$\begin{matrix} x+3y\geq6 \\ x+y \leq 10 \\ x \leq y \\ x\geq0, y\geq0 \end{matrix}$$

Find the point(s) of intersection

$$y=-\dfrac{1}{3}x+2$$

$$y=-x+10$$

$x=0:$

$$y=-\dfrac{1}{3}(0)+2, Point\ A(0,2)$$

$$y=-0+10, Point\ B(0,10)$$

$$-\dfrac{1}{3}x+2=x$$

$$\dfrac{4}{3}x=2$$

$$x=1.5, y=1.5,Point\ D(1.5,1.5)$$

$$-x+10=x$$

$$2x=10$$

$$x=5, y=5,Point\ C(5,5)$$

Point $A(0,2):𝑧(0,2) =3(0) + 9(2)=18$

Point $B(0,10):𝑧(0,10) = 3(0) +9(10)=90$

Point $C(5,5):𝑧(5,5) = 3(5) +9(5)=60$

Point $D(1.5,1.5):𝑧(1.5,1.5) = 3(1.5) +9(1.5)=18$

The function $z$ has a minimum with value of $18$ at $(0,2)$ and at $1.5,1.5).$