Hi everyone, I'm currently taking an abstract Algebra course and need a little guidance with a problem.

We are given two linear equations and need to solve for x and y using the method of "substitution" and again using "elimination". However, we must provide theorems and properties explaining each step. I have no problem solving the system, i'm just having difficulty citing the properties used.

Here is the system, which yields one solution (3, 2).

x + y = 5

x - y = 1

We first need to determine the appropriate algebraic structure in which we are solving the system: either a group, ring, field, or integral domain.

We were told that for solving something like x + 2 = 5 (with one variable) we would be in a "group" <Z,+> where + is the normal binary op of addition in Z.

Our text defines a "group" as having only one binary operation, while defining a "ring" as having two binary operations (addition and multiplication). We are then given the definitions of "field" and "integral domain" which seem to be special cases of a ring... So when solving the system x + y = 5 and x - y = 1 for x and y, would i then be in a "ring" since we have mult and addition? Would i be in the ring <Z,+,x> with the usual ops of addition and mult in Z? Or would i be in the reals? Or am i not even in a ring but rather a field or integral domain?

Next, we need to determine which properties and theorems i'll be using to solve the system. But since i can't determine whether we're in a group, ring, field or integral domain, i don't know which properties to cite.

We were told that when asked to solve for x in something like x + 2 = 5 we would use the theorem: For elements a and b in a group <G,*> if we are given that a=b, then for any c in G that a*c=b*c.

which when applied to solving for x, would let us do (x+2)+(-2)=5+(-2) which results in x+(2+(-2))=5+(-2) since addition is associative in G, which gives us x+0=3 since the additive ID in the group Z is 0, which gives us the answer x=3 using the definition of additive inverse and the definition of + in Z.

But how do i write something like this for solving the system of two linear equations with two variables x and y: x + y = 5 and x - y = 1?

I want to use the addition property of equality: (If a=b and c=d then a+c=b+d) but i'm not sure if this is of a group, ring, field, etc...

Can anyone tell me if i'm going in the right direction? Am i OK by saying i'd be in the ring <Z,+,x> with the usual ops of addition and mult in Z when solving for x and y here?

Thanks for the help!

## Bookmarks