- #1

Atran

- 93

- 1

Say I have the following equations:

y = x + 5

b = 2*a (the relation remains the same even if I change the variables)

Obviously, the solution is (x=a=5 and y=b=10) or (x=b=-5 and y=a=-10).

The first equation has the solution sets, A1={(x, x+5) : x∈R} and A2={(x+5, x) : x∈R}.

The second equation has the solution sets, B1={(x, 2*x) : x∈R} and B2={(2*x, x) : x∈R}

A1 ∩ B1 and A2 ∩ B2 are the solution sets to the system, if x=a and y=b.

A1 ∩ B2 and A2 ∩ B1 are the solution sets to the system, if x=b and y=a.

How can I prove that the variables which are meant to be equal (for example x=a), must be both the first or the second element in given pairs? For example if x=a and I consider A2, then I must consider B2, i.e. if x is the second element of the pairs x and y, then a must also be the second element of pairs a and b.

Am I thinking right? I've been looking at many websites but none really cleared up my confusion.

I'm really thankful if you can explain this clearly.