How do I convert a 3D line with 2 equals signs into something with 1 equals sign?

Question

I was given a line 2x+4 = 2z = 8y + 4

I am stumped by this because I don't know what to do with the two equals signs.

Subtracting 2z on the first and middle part will only lead to 2z canceling out when you try to make it all equal 0.

It almost looks like a parametric equation but even if it is I am stumped on converting it back into a equation with 1 equals sign.

Can you even make it have 1 equals sign? Thank-you!

Answer 1

This notation is just shorthand for a system of two equations, $$ 2z=2x+4,\qquad 2z=8y+4, $$ either of which has only one equals sign.

Answer 2

What you have are called "simultaneous equations" of a line.

$2x+4 = 2z = 8y + 4$

They are fequently witten like

$\frac {(x+2)}{4} = \frac {(y+\frac12)}{1} = \frac {z}{4}$ because the fractions actually make things easier.

First observation:

your line goes though the point $(-2,-\frac 12,0)$

Notice that if $(x,y,z)$ equals the values above, then $0=0=0$

And if we increase $x$ by $4$, $y$ by $1$ and $z$ by $4$

$(x,y,z) = (2,\frac 12, 4)$ the equations balance.

Parmateric form is another handy way to write a line

$x+2 = 4t\\ y + \frac 12 = t\\ z = 4t$

we can clean that up a little bit:

$x = 4t-2\\ y = t- \frac 12\\ z = 4t$

And the next most common way to represent the line is vector form.

$(x,y,z) = (-2,-\frac 12,0) + (4,1,4)t$

Hope this helps.