Where lines meet, and the foot of a perpendicular

5 minAS Level1.3.11.3.2

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can find where two lines cross, find the foot of the perpendicular from a point to a line, and use it to get the shortest distance from the point to the line.

Meeting point = shared solution

A point lies on a line exactly when its coordinates fit the line's equation. So the point where two lines cross is the one $(x, y)$ that fits both equations at once.

That's a pair of simultaneous equations:

Both in the form $y = \dots$ ? Set them equal.
Otherwise, substitute or eliminate, as you did at IGCSE.

Worked example: Finding the crossing point

Find the point of intersection of $2x + y = 7$ and $x - 3y = 7$ .

Have a go first?

When there's no crossing

Two lines with the same gradient are parallel.

If they're different lines, they never meet. The simultaneous equations then have no solution: the algebra collapses into nonsense like $0 = 5$ .

Gradients	Number of meeting points
different	exactly one
equal, different intercepts	none (parallel)
equal, same intercept	infinitely many (same line)

Quick check

CalculateQuestion 1 · Recall

CalculateQuestion 2 · Apply

Answer the 3 questions above to continue.