Where Two Lines Meet

Two equations together are a system. A solution must make both true at once.

On a graph

Each equation is a line. The point where they cross satisfies both.

By substitution

If y = 2x and y = x + 3, then 2x = x + 3, so x = 3 and y = 6.

When there is no crossing

Parallel lines have the same slope and never meet, so the system has no solution.

A solution satisfies both equations

A system of two equations asks for values that make both true at once. Graphically, that is the point where the two lines cross — the one place that lies on both.

Three possibilities again

Two lines can cross once, be parallel and never cross, or be the same line and cross everywhere. So a system has one solution, none, or infinitely many — mirroring the single-equation cases exactly.

Graphing is intuitive but imprecise

Reading an intersection off a graph gives an approximate answer, and if the solution is not at a grid point you cannot read it exactly. Graphing shows you what is happening; algebra gives you the number.

Check in both equations

A proposed solution must satisfy both equations. Checking only one is not a check at all, since any point on one line satisfies that equation. This is the most commonly skipped step in the topic.