Planes through line parallel to axes

Given line. The line of intersection of 2x + 3y − 5z = 4 and 3x − 4y + 5z = 6.

Any plane through it is

(2x + 3y − 5z − 4) + λ(3x − 4y + 5z − 6) = 0,

that is,

(2 + 3λ)x + (3 − 4λ)y + (−5 + 5λ)z − (4 + 6λ) = 0.

A plane is parallel to a coordinate axis when the coefficient of that variable is zero.

Parallel to the x-axis

2 + 3λ = 0, λ = −2/3. Then the y-coefficient is 3 + 8/3 = 17/3, the z-coefficient is −5 − 10/3 = −25/3 and the constant is −(4 − 4) = 0.

17y − 25z = 0

Parallel to the y-axis

3 − 4λ = 0, λ = 3/4. The x-coefficient is 2 + 9/4 = 17/4, the z-coefficient is −5 + 15/4 = −5/4 and the constant is −(4 + 9/2) = −17/2.

(17/4)x − (5/4)z − 17/2 = 0, so 17x − 5z − 34 = 0

Parallel to the z-axis

−5 + 5λ = 0, λ = 1. The x-coefficient is 5, the y-coefficient is −1 and the constant is −10.

5x − y − 10 = 0

Each plane passes through the given line; for example, adding the two given planes (λ = 1) gives 5x − y = 10, which does not contain z.