What a System Is, and What It Means to Solve One

Two demands at once

A single linear equation in x and y, like x + y = 5, has many solutions: (0,5), (2,3), (5,0), and infinitely more. Each one is an ordered pair that makes the equation true. A system of equations — also called simultaneous equations — lines up two or more such equations and asks for a pair that satisfies all of them at once.

So a solution of a system is stricter than a solution of one equation. It is not enough to fit the first equation; the same numbers must also fit the second. Picture each equation as a line in the coordinate plane: the solution is the point where the lines cross.

Checking a candidate honestly

Before learning any solving method, learn to verify. Given a pair, check the solution by substituting it into every equation and confirming each one balances. If it fails even one, it is not a solution of the system, no matter how nicely it fits the rest.

System:   x + y = 5
          x - y = 1

Claim: (3, 2) is the solution.

Check equation 1:  3 + 2 = 5   ✓ true
Check equation 2:  3 - 2 = 1   ✓ true

Both hold, so (3, 2) is the solution of the system.

Counter-check: is (4, 1) a solution?
  Eq 1:  4 + 1 = 5   ✓
  Eq 2:  4 - 1 = 3 ≠ 1   ✗
(4, 1) fits the first line but misses the second — rejected.

1. Write the candidate pair as (x, y), naming which number is x and which is y.
2. Substitute into the first equation and simplify both sides.
3. Substitute the same pair into every remaining equation.
4. Only call it a solution if every check turns into a true statement.