<s>[INST] Alternative methods for solving a system of one linear one non linear simultaneous equations Take the equations $$x+y=5$$ $$x^2 + y^2 =13$$
The most basic method to solve this system is to first express the linear equation in terms of one of the variables and then sub that into the non-linear equation.
But I am curious if there are other methods to solve such a system ?
 [/INST] The 'ac' method can be used.
From above:
2x^2 + (-10x) + 12 = 0
a = 2, c = 12
a*c = 24
The possible factors of 24 are: (24*1),(12*2),(6*4),(3*8)
In the 'ac method' the sets of factors must multiply to 24 and sum to -10.
By examination we find that this is true of only one of the above factors, (6*4) where (-6*-4) = 24 and (-6 + -4) = -10.
2x^2 + (-10x) + 12 = 0
but, as we found, -10x = (-6x + -4x) so by substitution..
2x^2 + (-6x + -4x) + 12 = 0
Regrouping we get..
[2x^2 - 6x] + [-4x + 12] = 0
Factoring like terms out we get..
2x(x-3) + -4(x-3) = 0
Factoring (x-3) out we get..
(x-3)(2x-4) = 0
The zeros occur where x-3 = 0 and where 2x-4 = 0
x-3 = 0, x = 3 
2x - 4 = 0, 2x = 4, x = 4/2, x = 2
Given x+y = 5
when x = 3, 3+y=5, y = 5-3, y = 2
when x = 2, 2+y=5, y = 5-2, y = 3
Answers:
x = 3, y = 2 and x = 2, y = 3
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