How Would You Say: ${(x,y,z)|5x+2y+z=3}$ Currently studying systems of linear equations, and came across this statement: 
$\{(x,y,z)\, |\, 5x+2y+z=3\}$ 
I know what what the statement means, I understand that it involves set notation, I just don't know how to 
articulate statements of set notation when it involves ordered-pairs/ordered-triples. 
Can someone please help me articulate this statement. Thank you.
 A: This is the set of all ordered triples $(x,y,z)$ such that $5x+2y+z=3$.
A: The meaning of that can be thought of every point in space, $(x,y,z)$, such that they should solve the equation $5x+2y+z=3$. So for example:
$$
A = \left(1,\frac{1}{2},-3\right)
$$
where
$$
x=1\text{ , }
y=\frac{1}{2}\text{ , }
z=-3
$$
and is an ordered-triple that solves your equation, so that point is a point in the "solution world", because:
$$
5(1) + 2\left(\frac{1}{2}\right) (-3) = 3
$$
Another example is the origin $(0,0,0)$. It isn't in the "solution world". When I say "solution world", I mean the set of all ordered-triples that solves your equation.
I don't know if you know about planes, but $5x+2y+z = 3$ can be seen as a plane in $\mathbb{R}^3$, because when you plot all the points in the canonical coordinate system, you'll actually get a plane and see that as a geometric interpretation of the "solution world".

It's important to say that what I've just said is a geometric interpretation of what you were asking. When you go above triples, you are "kinda" losing it but everything still holds.
Hope it helps!
