Q.If $4x + i(3x - y) = 3 + i(-6)$, where $x$ and $y$ are real numbers, then find the values of $x$ and $y$.

Ordered Pair Equality: From Intuition to Precision

Think about a simple list of two things — say, your name and your age. If I write (Ravi, 15), that's an ordered pair. The word "ordered" is the key: the first position and the second position mean different things. (Ravi, 15) is not the same as (15, Ravi), because the first one tells you a name first, the second tells you an age first.

Now, when are two such pairs equal? Intuitively, they are equal only when both the first things match and both the second things match, in that exact order.

So (Ravi, 15) equals (Ravi, 15), but it does not equal (15, Ravi) — even though both contain the same two items. The order matters.

The Precise Statement

Read this as: "The ordered pair (a, b) equals the ordered pair (c, d) if and only if a equals c and b equals d."

Two conditions must hold simultaneously:

1. The first components are equal: $a = c$
2. The second components are equal: $b = d$

If either condition fails, the pairs are different.

Why This Matters

This definition is the foundation for everything that uses ordered pairs — coordinates in the plane, relations, functions, and even complex numbers. When you plot the point $(3, 5)$ on a graph, you are implicitly using this rule: $(3, 5)$ is a different point from $(5, 3)$ because the first coordinates differ.

Quick Check

Which of these are true?

• $(4, 7) = (4, 7)$ → True (both components match)
• $(4, 7) = (7, 4)$ → False (first components differ: $4 \neq 7$)
• $(x, 5) = (3, 5)$ → True only if $x = 3$
• $(p, q) = (q, p)$ → True only if $p = q$

The last one surprises many students. If $p = q$, then the pair becomes $(p, p)$ and swapping gives the same thing. But if $p \neq q$, they are different.

One More Layer: Why "Ordered"? …