Q.Find the value of the following: Minimise $Z = 3x + 5y$ such that $x + 3y \ge 3$, $x + y \ge 2$, $x, y \ge 0$.

Minimise $Z=3x+5y$ over the region $x+3y\ge3,\ x+y\ge2,\ x,y\ge0$.

Feasible corner points:

• $x+3y=3$ meets $x+y=2$ at $(1.5,\ 0.5)$.
• $x+3y=3$ meets the $x$-axis at $(3,0)$.
• $x+y=2$ meets the $y$-axis at $(0,2)$.

Evaluate: $Z(1.5,0.5)=4.5+2.5=7,\quad Z(3,0)=9,\quad Z(0,2)=10.$ …

Set up

We minimise $Z=3x+5y$ subject to

Both slanted constraints are "$\ge$", so the feasible region lies above the two lines in the first quadrant — it stretches off to infinity (unbounded).

Find the corner points

1. $x+3y=3$ and $x+y=2$. Subtracting, $2y=1$, so $y=\tfrac12$ and $x=\tfrac32$. Corner $(1.5,\ 0.5)$.
2. $x+3y=3$ with $y=0$: $x=3$. Corner $(3,0)$ — check $x+y=3\ge2$ ✓.
3. $x+y=2$ with $x=0$: $y=2$. Corner $(0,2)$ — check $x+3y=6\ge3$ ✓.

(The points $(0,1)$ and $(2,0)$ each lie on one line but fail the other constraint, so they are not corners of the region.)

Evaluate the objective

The smallest corner value is $7$. …

Method: Minimising over an unbounded feasible region ($\ge$ constraints)

With "$\ge$"-type constraints the feasible region usually opens to infinity, so the corner-point value must survive an extra check before you can call it the minimum.

Steps

Step 1: Plot the lines and shade the "$\ge$" side.

For each $ax+by\ge c$, the feasible side is away from the origin (test $(0,0)$). Combined with $x,y\ge0$, the region typically stretches up and to the right — an unbounded region.

Step 2: Find the corners and evaluate.

Keep only intersection points satisfying every constraint, and compute $Z$ at each; call the smallest value $M$. …

Common Mistakes

Mistake 1: Treating a "$\ge$" region as bounded and stopping at the smallest corner.

Why it's wrong: the region is unbounded, so a corner value is the minimum only after a check. Correct approach: confirm $3x+5y<7$ shares no point with the region, which proves $7$ is the true minimum.

Mistake 2: Taking an intercept like $(3,0)$ or $(0,1)$ as a corner without testing both constraints. …