As title. My colleague said yes and I just can't understand.
I tried inputting some points into this site to show as long as the coordinates satisfy the same equation, the points are coplanar. But I don't know why she said the equation just depicts a relationship btw the coordinates and don't decide if the points are on the same plane.
To be more specific, the equation is something like $f(x, y, z) = g(y, z) = 0$ and the $x$ coordinate does not play a role. And the y, z coordinates of all the points I considered satisfy the equation $g(y, z) = 0$.
Edit: case closed. Looks like she finally grab the sense of where the points are at and agreed that those points are coplanar. But thanks for both answers and I'm leaving for the audience to decide which one is better.
Assuming 3D, no. You are correct that if points satisfy the equation of a plane, then it will be on that plane (thus coplanar).
If you have points and a plane, but you only look at 2 out of 3 dimensions, then it is possible the points satisfy 2 out of 3 dimensions but are not on the plane. If you consider all 3 dimensions then it is not possible to have a point not on the plane satisfy the equation of the plane.
(All assuming ordinary 3d geometry)
EDIT: I should not write answers right before I fall asleep!
What I was trying to get at with my above answer is that if there was a plane, the $xy$ plane for example, there could be points with a $z \neq 0$ that appear to be on the plane if you ignore the $z$ dimension. However, these points are actually not on the plane since the equation for the $xy$ plane includes the criteria that $z=0$.
Perhaps you can explain it to your colleague in a different way that may be more intuitive to them. For example, let's focus on 2D, only looking at $x$ and $y$. Take the equation of a line, say $x=y$, and plot all of the points that make up that line. For example, start with $x,y=0$. Then add $x,y=1$. Add $x,y=-1$, and so on. Now, there will still be some space in between. Add $x,y=0.5$, $x,y=-0.5$, and so on. Now you see there's less "space" in between the points. You can keep adding an infinite amount of points, and then you will have the complete line, $x=y$.
This shows that the line $x=y$ is only made up of points that satisfy that equation. Now you can modify this example for a plane, and show that a plane is made up of literally every point that satisfies the equation of the plane. Thus, every point that satisfies the equation of a plane will be on the plane.