If x=y in an equation, are there an infinite number of solutions?

Question

The problem is actually...

x'=xcos(alpha)+ysin(alpha)
y'=-xsin(alpha)+ycos(alpha)
x=x'cos(alpha)-y'sin(alpha)
y=x'sin(alpha)+y'cos(alpha)

Answer 1

Not necessarily. I assume x' and y' are just different variables (maybe try writing these with different letters like w and z to avoid confusion).



btw, I haven't gone through this problem, I'm just giving a hint to help you out.



If everything becomes equivalent to x=x (or y=y, etc.) then the equations are ALWAYS true for all x, y, x', y', because x=x is always true no matter what x is (same for y=y, etc.).



If you end up with x = y, you still have to do more work. You can try substitution, but there's also a chance you might end up with a system of equations that you have to solve (which I assume you can do if you're in or past algebra 2).



Wait... just to make things clear, you meant when x=y in a "system of equations" like the one you have right? Of course the "equation" x = y has an infinite number of solutions - just plug any number in for x (then it's y also).



btw, here's a quick example showing that a system reducing to x = y might not have an infinite number of solutions:



x= 6 - y

y = 4 - 1



Then x = y = 3. One solution.

Answer 2

x=y is a straight line on which finite number of combination of x,y are available hence the solution set for x=y is:

{(x,y): x∈R and y∈R}

Answer 3

yes