How can $2x -5y +z = 3$ be equation of a line?

Question

The equation of the plane containing the lines $2x - 5y + z = 3$, $x + y + 4z = 5$ and parallel to the plane $x +3y + 6z = 7$? I actually have the solution to this question, but I don't understand it. I am a bit new to 3d geometry. Isn't equation of a plane always of form $ax+by+cz=d$? Aren't the two lines mentioned in the question actually planes? The solution I have uses the two equations for lines as equations for planes in family of planes (i.e., $P_1 +kP_2=0$). Basically my doubt is: why are $\boldsymbol{2x - 5y + z = 3}$ and $\boldsymbol{x + y + 4z = 5}$ called lines when they are obviously equations of planes? Is there something I'm missing here?

Answer 1

Actually they are referring to the line formed by intersection of the two planes$ 2x−5y+z=3$and $x+y+4z=5$. There is mistake in understanding the question

Answer 2

They should not be called lines, but planes. You are absolutely right. I am pretty sure that they made a mistake.

Answer 3

Maybe the start of the question said the line (singular, not plural) determined by $2x - 5y + z = 3, x + y + 4z = 5$, which is correct because it denotes the intersection (you want both equations to hold simultaneously) of two non-parallel planes in $\mathbb{R}^3$, so these determine a unique line.

You are right that one equation $ax+by+cz+d$ determines a plane in $\mathbb{R}^d$ but two of these equations that hold simultaneously typically determine a line (or the set is empty, as in two parallel planes, or a plane when the two equations denote the same plane). This is only true in $\mathbb{R}^3$..

Answer 4

The only way I can see of making sense of this question is to read it as:

"[Find the] equation of the plane containing the line of intersection of the planes $\ 2x - 5y + z = 3\ $ and $\ x + y + 4z = 5\ $, and parallel to the plane $\ x +3y + 6z = 7\ $."

Since the line of intersection of the first two planes happens to be parallel to the third, this problem has a unique solution. It doesn't help to merely replace the word "lines" with "planes" in the original statement, because the expression "the equation of the plane containing the planes $\ 2x - 5y + z = 3\ $ and $\ x + y + 4z = 5\ $, ..." doesn't make any sense. No such plane exists.