For what value(s) of $\displaystyle m$ will the system of equations,
$\displaystyle x+my=2$
$\displaystyle (m-1)x+2y=m$
have:
a) a unique solution?
b) no solution?
c) an infinite number of solutions?
Try solving it! For example, you can multiply the first equation by 2 and the second equation by m and subtract to eliminate y:
$\displaystyle (2x+ 2my)- (m(m-1)x+ 2my)= 4- m^2$
$\displaystyle (2- m^2+ m)x= 4- m^2$
$\displaystyle x= \frac{m^2- 4}{m^2- m- 2}$
Now, if the denominator is not 0, there is a unique solution. If the denominator is 0 and the numerator is not, there is no solution. If both numerator and denominator are 0, there are an infinite number of solutions.