Hi everyone, i'm hoping to get a little confirmation as to whether or not i'm on the right track with something.
I've been given two linear equations and need to solve for x and y using the method of "substitution" and again using "elimination". However, i must also:
1. State the algebraic structure in which to solve the system (ie: group, ring, field, integral domain). 2. Once i have chosen the algebraic structure, i must cite theorems/properties for EACH step of the solutions.
Here is the system, which yields one solution (3, 2).
x + y = 5 x  y = 1
Here's my work for solving the system in the two methods:
*************************************************************** For the "elimination" method:
x+y=5 xy=1
1(1x+1y)=(5)1 Multiply the top equation (both sides) by 1 1(1x11y)=(1)(1) Multiply the bottom equation (both sides) by 1
note 1+(1)=0
add eqns together
(1x1x)+(1y+1y)=51 (101)x+(1+1)y=51 1+(1)x+(1+1)y=51 0+(1+1)y=51 2y=4
divide both sides by 2... 2y/2=4/2 y=2
plug into eqn x+y=5 1x+1(2)=5 1x=52 subtract 2 from both sides 1x=3 (1/1)(1)x=(3)(1/1) mult both sides by (1/1) x=3
So solution is (3,2)
********************************************
By the "substitution" method:
x+y=5 xy=1
solve for y in eq
1y=51x Subtract 1x from both sides y=(51x) Divide both sides by 1 y=51x
substitute
1x+(1)(51x)=1 1x1(5)1(1)x=1 Distribute 1 to 51x 1x5+1x=1
1x5+1x+5=1+5 Add 5 to both sides 1x+1x=6 2x=6
(1/2)(2/1)x=(6/1)(1/2) Mult both sides by (1/2) x=3
Substitute into eq
1(3)1y=1 31y=1 1y=13 Subtract both sides by 3 1y=2 (1/(1))(1)y=(2/1)(1/(1)) Mult both sides by 1/(1) y=2/(1) y=2
So solution is yet again (3,2)
**********************************************
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.
But since (as my work shows above for each method of solving) i now have two binary operations (and rational numbers) so i cannot solve the system in a "group" of integers. So when solving the system x + y = 5 and x  y = 1 for x and y, would i then be in a field (of rationals) since i have two binary operations and fractions?
Once i know what structure i'm working in, i can then cite specific properties and theorems from that structure at each step of my work. Note i need to do this twice; once for the "elimination" method and once for the "substitution" method.
The start of my work needs to start with a line that looks like:
To solve this system of linear equations, i will be in the (group, ring, field, integral domain) represented by <Z,R,Q,+,x,?,?> where + and x are the ordinary binary ops of the (integers, real numbers, rational numbers).
What would this line look like?
Then once the structure is decided, i would cite a theorem from that structure that looks like:
For any a and b in a (group, ring, field, integral domain) <Z,R,Q,+,x,?,?> if we know a=b, then for any c we know that a*c=b*c...
Can anyone tell me if i'm on the right track? I'm thinking field of rational numbers, but i'm not sure if that's the easiest... Thanks in advance!
