Find the value of the following:

Maximise $Z = x + y$, subject to $x - y \le -1$, $-x + y \le 0$, $x, y \ge 0$.

Question

Find the value of the following:

Maximise $Z = x + y$, subject to $x - y \le -1$, $-x + y \le 0$, $x, y \ge 0$.

Accepted Answer

> [!TLDR]
> The constraints $x - y \le -1$ and $-x + y \le 0$ together with non-negativity create an **infeasible region** — no point satisfies all conditions simultaneously. Hence, the maximum does not exist; the problem has **no feasible solution**.

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## Why This Happens — The Core Idea

In Linear Programming, the first thing we always check is whether the constraints actually define a region where all conditions hold at once. If they don't, there's nothing to maximise. This problem is a classic trap: the constraints look simple, but they contradict each other when you combine them with $x, y \ge 0$.

Let’s see why.

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## Step-by-Step Reasoning

**1. Rewrite the constraints in a clearer form.**

We have:
- $x - y \le -1$  →  $y \ge x + 1$
- $-x + y \le 0$  →  $y \le x$
- $x \ge 0$, $y \ge 0$

So the first constraint says $y$ must be at least $x + 1$. The second says $y$ must be at most $x$.

**2. Can both hold at the same time?**

If $y \ge x + 1$ and $y \le x$, then we need:
$$x + 1 \le y \le x$$
which implies $x + 1 \le x$, i.e. $1 \le 0$. That’s impossible.

No matter what $x$ and $y$ are, these two inequalities cannot be satisfied together.

> [!WARNING]
> A common mistake is to graph each inequality separately and look for an overlapping region — but here the overlap is empty. Don’t assume a solution exists just because each inequality individually has solutions.

**3. What about the non-negativity constraints?** …

Type	Symbol	Meaning
Upper bound	$\leq$	cannot exceed a limit (e.g. labour $\leq 300$ )
Lower bound	$\geq$	must meet a minimum (e.g. protein $\geq 50$ )
Non-negativity	$x,\,y \geq 0$	quantities cannot be negative

Q.Find the value of the following: Maximise $Z = x + y$ , subject to $x - y \le -1$ , $-x + y \le 0$ , $x, y \ge 0$ .

Concept understanding — Linear Programming Constraints

Linear Programming Constraints

What a constraint looks like

Types you will meet

How constraints shape the problem

Why This Happens — The Core Idea

Step-by-Step Reasoning

Method: Graphical Method — Test Feasibility Before Optimising

Steps

Common Mistakes

Q.Find the value of the following: Maximise Z=x+yZ = x + yZ=x+y, subject to x−y≤−1x - y \le -1x−y≤−1, −x+y≤0-x + y \le 0−x+y≤0, x,y≥0x, y \ge 0x,y≥0.

Why This Happens — The Core Idea

Step-by-Step Reasoning

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Q.Find the value of the following: Maximise $Z = x + y$ , subject to $x - y \le -1$ , $-x + y \le 0$ , $x, y \ge 0$ .

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