What is a system of two equations?

Definition

A system of two equations with two unknowns (often denoted x and y) is a set of two equations that must be true at the same time. It is written using curly brackets:

{ 2x + y = 7
{ x - y = 2

Solving this system means finding the pair or pairs of (x, y) that satisfy both equations simultaneously.

Graphical interpretation

Each equation of the form ax + by = c represents a straight line in a coordinate system. Solving the system amounts to finding the point of intersection of two straight lines.

Case	Number of solutions	Graphical interpretation
Secant lines	1 solution	A single point of intersection
Distinct parallel lines	0 solutions	No common points
Coincident lines	An infinite number of solutions	The two lines coincide

Example

For the system above, we can verify that (3, 1) is a solution:

• 2(3) + 1 = 7 -> true
• 3 - 1 = 2 -> true

Both equations are satisfied, so (3, 1) is indeed THE solution to the system.

Common pitfall

Note: a solution that satisfies only ONE of the two equations is not a solution to the system. You must always check both equations; never just one.