I don't know if that type of equation has a name, but it would have an infinite number of solutions, as you more or less said.
If it was the result of eliminating a variable while solving a system of equations, that system would be called dependent
BTW, an equation cannot equal another equation: what you meant to say is that the equation
) becomes the equation
when you try to solve for
. I know what you meant, but there are some here that have little tolerance for young or inexperienced posters who don't quite get the terminolgy correct. Kind of like the Grammar Police.