the intersection of curves

Where do two curves cross? A point lies on a curve exactly when its coordinates satisfy that curve's equation, so a point lies on both curves exactly when its coordinates satisfy both equations at once. Finding the intersection is therefore the same task as solving a system of equations simultaneously — the geometry of 'crossing' is the algebra of 'solving together'.

The standard method is substitution: solve one equation for one variable and substitute into the other, reducing two equations in x and y to a single equation in one variable, which you then solve. The number of real solutions is the number of crossing points. For two straight lines this leads to one solution (they meet once), no solution (parallel), or infinitely many (the same line twice). For a line and a circle, substitution produces a quadratic, whose two, one, or zero real roots mean the line is a secant (cuts the circle), a tangent (just touches), or a misser (no contact) — matching what you would see geometrically.

This is one of analytic geometry's most practical powers: any question of the form 'do these shapes meet, and where?' becomes routine algebra. Two honest cautions. First, count solutions with care — a repeated (double) root signals a tangency, a single touching point, not two separate crossings. Second, when you square an equation along the way to remove a square root, you can manufacture extraneous solutions that solve the squared system but not the original; always substitute candidate points back into both original equations to confirm them.