No.
No.
That system of equations has no solution. When the two equations are graphed, they turn out to be the same straight line, so there's no such thing as a single point where the two lines intersect. There are an infinite number of points that satisfy both equations.
x=4y+1 x=4y-1 No,they have different solutions.
The two equations represent the same straight line.
Each of these have an infinite number of solutions. Never can they be equal.
No, equations with the same slope do not intersect unless they are the same line.
I notice that the ratio of the y-coefficient to the x-coefficient is the same in both equations. I think that's enough to tell me that their graphs are parallel. So they don't intersect, and viewed as a pair of simultaneous equations, they have no solution.
Equivalent equations
They are simultaneous equations
Equivalent equations are equations that have the same solution set.
Indeed, these two equations are the same - because addition is a commutative function.
There is no special name. Two totally unrelated equations could have the same solution(s).