How Do You Find the Intersection of Two Planes or Three Planes?

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Subtracting the second equation from the first gives $2y=2$, so $y=1$. Substitution gives $x+z=2$. Let $z=t$; then $x=2-t$, so the line of intersection is

The direction vector can also be found from the cross product of the planes' normal vectors.

Before solving, compare the normal vectors. If they are not scalar multiples, the planes intersect in a line. If they are scalar multiples, compare the constants: proportional equations describe coincident planes, while inconsistent constants describe distinct parallel planes. This check predicts the geometry and helps verify your algebra.

Intersecting Three Planes

Add the third plane's equation and solve the resulting system by elimination or row reduction. For example, if $P_3:x-z=0$, substituting $x=2-t$ and $z=t$ gives $2-t-t=0$, hence $t=1$. Therefore, all three planes meet at $(1,1,1)$.

A common misconception is that two planes normally intersect at a point. Unless they are parallel or coincident, two planes intersect along a line.

Exam Technique

For HL Paper 1, show elimination and write the final point or vector equation clearly. On Paper 2, row reduction using a GDC is appropriate, but include the system or augmented matrix and interpret free variables or contradictions; calculator output alone is insufficient.