<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 quadratic equation can be used.
Given:
x + y = 5, then y = 5 -4

Given
x^2 + y^2 = 13
then x^2 + (4-x)^2 = 13
and x^2 + x^2 - 10x + 25 -13 = 0
2x^2 + (-10x) + 12 = 0

Then the co-factors are a = 2, b = -10, c = 12
y = [-b (+-) sqrt(b^2 - 4ac)]/[2a]  <-- Quadratic Formula
y = [-(-10) (+-) sqrt((-10)^2 - 4(2*12))]/(2*2)
y = [10 (+-) sqrt(100-96)]/4
y = [10 + 2]/4  and  y = [10-2]/4
y = 12/4  and  y = 8/4
y = 3  and  y = 2
given x + y = 5
When y = 3, x + 3 = 5, x = 5-3, x = 2
when y = 2, x+2 = 5, x = 5-2, x = 3
Answers: x = 3, y = 2 and x = 2, y = 3
Try your answers in all of the original equations and against any given or implied restrictions to make sure they work. They do! (Always check for 'extraneous' answers.)
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