Where Two Lines Meet

You'll learnto find the one point that lies on both lines, then report the exact quantity asked.

A system asks for the one point that lies on both lines at once. Solve it by substitution or elimination, read whether the lines cross once, never, or everywhere — and then hand back the exact quantity asked, which is not always a single variable. Watch the sign that vanishes mid-substitution and the answer that stops one step short.

8 STEPS · 6 QUICK CHECKS · the intersection is the solution · answer what is asked · v3

two lines, one point

Before you start

What you're looking at

A coordinate plane with two lines, y = 2x − 1 and 3x + y = 9, meeting at the green point (2, 3).

The question

Where do the two lines cross — the single point (x, y) that satisfies both equations at once?

Watch for

The lines grow outward until they intersect; watch the green point appear where both lines pass through the same place.

What the symbols mean

Each colored line is one equation; the legend names them, and the green point marks the shared solution that lies on both.

You should already know

How to graph a line from its equation (Topic 1.1) — every point on a line makes that line’s equation true, which is why the crossing makes both true.

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