The problem is to find the values of x and y that solve the following system of linear equations:
6751x + 3249y = 26751
3249x + 6751y = 23249
However, the caveat is that I must contrive a mental process of solving them--I must find x and y in my head.
The Attempt at a Solution
Rather than waste time, I figured I would analytically solve for x and y, to see if the solved values elucidated some pattern; and then I would contrive a way to solve this problem mentally. I found that x = 3 and y = 2. This didn't seem to be of much advantage.
Before I did this, however, I jotted down a few initial observations:
(1) Notice, the coefficients of the different variables in different equations have the same
value--6751x from the first equation and 6751y from the second equation. So, whatever value is chosen for x must different for y (I am not certain of my reasons of this observation. So, if someone could help me work through this observation, I would appreciate it.).
(2) I also noticed that the RHS of equation 1 is 20000 plus the coefficient of x. A Similar observation can be acquired by looking at equation 2.
I would appreciate someone who could stimulate my thoughts, but please don't just give me the solution. I want to be able to solve this problem on my own, to some degree, of course.