Finding where a line intersects a plane

Problem
Where does the line r = (1, 2, -5) + t (2, -3, 1) meet the plane 2x + 5y - 3z = 6?

Solution
Letting r = (x, y, z) we have

(x, y, z) = (1, 2, -5) + t (2, -3, 1). For points on this line, the components of their position vectors are given by
x = 1 + 2t
y = 2 - 3t
z = -5 + t Also, we know that 2x + 5y - 3z = 6 for points lying in the plane.
For a point in common, all these equations must be simultaneously satisfied. Substituting for x, y and z in the 4th equation gives
2 (1 + 2t) + 5 (2 - 3t) - 3 (-5 + t) = 6. This gives -14t = -21 so t = 3/2. Substituting this value of t gives x = 4, y = -5/2 and z = -7/2.
The position vector of the point where the line meets the plane is ( 4, -5/2, -7/2).
Check for yourself that this point does lie on the plane 2x + 5y - 3z = 6. (Doing this gives a quick check against arithmetical slips!)

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