# Thread: Variables within variables? Simultaneous equation solutions

1. ## Variables within variables? Simultaneous equation solutions

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:

$2x-2y=5$

$10x-10y=m$

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 ?

$10x-10y=10y$

$y = 0$ (sub into first equation)

Am I way off track with this question?

$2x = 5$

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

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

2. ## Re: Variables within variables? Simultaneous equation solutions

No, you're not off track. The only way there are infinite solutions for real values of $m$ are when $m=25$, and the only way there are no solutions are when $m\neq{25}$

I can't see any way to find real values of m that give 'just one solution'.