|A set of simultaneous equations, such as:
2x + 3y = 12 
5x - 6y = 3 
can be solved by methods of elimination, substitution, matrixes.
Investigate a fourth way - iterative guessing.
Equations  and  can be written:
x = 6 - 3/2y
y = 5/6x - 1/2
Suppose we start with a guess of x = 0 and y = 0. This gives the new values of:
x = 6
y = -0.5
Are these new values better guesses? Try... this is known as iteration.
x = 6.76
y = 4.5
Try with new:
x = -0.75
y = 5.125
These equations appear to go nowhere. A different re-arrangement allows a different iteration:
y = 4 - 2/3x
x = 3/5 + 6/5y
If the iterations start with x = 0 and y = 0:
After 60 iterations, you end up with x = 3.004 and y = 2.003 which is close to the correct answer
(x = 3, y = 2)
Why did the first attempt fail and the second attempt succeed (albeit slowly)?
Consider this pair of equations:
x - 2y = -2
2x + y = 2
y = 1/2(x + 2)
x = 1/2(2 - y)
After 8 iterations you get x = .398 and y = 1.195 (close enough to x = .4 and y = 1.2).
Why did it work so much better for this than the first set?
Investigate this method with the aim of coming up with a rule that will enable you, given a pair of sim equations:
ax + by = c
dx + ey = f
to determine which way will converge to a true solution.
Can you derive any rules that will enable you to predict how rapidly the iterative scheme will converge?
I'm at a loss. Anyone know or have ideas to try out?