Solving Linear Systems (Basic Question)

[Atran]

Hi. I don't understand how a solution to a linear system is obtained (for example geometrically; don't consider the substitution method and elimination), and I am feeling very frustrated.

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.

 

[Mark44]

Hi. I don't understand how a solution to a linear system is obtained (for example geometrically; don't consider the substitution method and elimination), and I am feeling very frustrated.

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 two equations involve different pairs of variables, so the two equations aren't related at all. If the second equation happened to be y = 2x, then you would have two lines that intersect. The intersection point would be the solution of the system of equations.

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}

By convention we write the ordered pairs as (x, y), in that order. The solution set would be {(x, y) | y = x + 5, x $\in$ R}, and similar for the second equation.

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.

Why are you doing this? By using a different set of variables (a and b), you are overcomplicating what is a simple problem.

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.

You don't need to prove this, as the variables appear in a certain order by convention. If you write the system as
y = x + 5
y = 2x

then the system is trivial to solve.

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.

 

[1MileCrash]

I don't understand how those two equations form a linear system, or your "solution."