Question

How many solutions does the system formed by x - y = 3 and ay + 3a = 0 have for a non zero number a? Explain and show how you get the answer please

Answer 1

Solve both equations for y:
$x-y=3 \ \ \ |-x \\ -y=-x+3 \ \ \ |\times (-1) \\ y=x-3 \\ \\ ay+3a=0 \ \ \ |-3a \\ ay=-3a \ \ \ |\div a, a \not= 0 \\ y=-3$

Set the expressions equal to each other:
$y=y \\ x-3=-3 \ \ \ |+3 \\ x=0 \\ \\ (x,y)=(0,-3)$

So, for every nonzero number a the system of equations has one solution.