I have been asked to find real values for m such that this system of equations has no solution, infinitely many solutions and exactly one solution:

$\displaystyle 2x-2y=5$

$\displaystyle 10x-10y=m$

At first I thought that the question wanted me to find a value for each scenario, but I could only find a value that gave infinitely many solutions: m = 25. I can see that they are essentially the same line so any value other than 25 would just shift it up or down and cause the two lines to never intersect.

I couldn't see any other real value for m could produce any solutions until I though what if m = 10y ?

$\displaystyle 10x-10y=10y$

$\displaystyle y = 0$ (sub into first equation)

Am I way off track with this question?

$\displaystyle 2x = 5$

$\displaystyle x = \frac{5}{2}$

Does that count? I guess there would be infinitely many ways to do that though...