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x + y = 7 => y = 7-x

- cereskara December 10, 2014Plug into second equation

x^2 + (7-x)^2 = 29

=> x^2 + 49 -14x +x^2 = 29

=> 2x^2 -14x + 20 = 0

Use quadratic equation/ complete the square or whatever you want. I'll do quadratic.

(14 +- sqrt(14^2 - 4(2)(20))) / 2(2)

=> x = 2, 5

Plug it back into one of the equations of your choice, I'll use the first

x = 2

2 + y = 7

y = 5

x = 5

5 + y = 7

y = 2

Our two pair of solutions are (x=2, y=5) and (x=5, y=2)

Edit: On the generalizing part, what you're talking about are systems of equations. What the solutions mean are basically what pairs points of x and y satisfy all of the equations.

Visually, what this means is if you graph them, at what points do they all intersect?

The set of solutions can be finite - they intersect at some points, infinite - they overlap each other so intersect at all their points, empty - they do not intersect.

What you're also asking about for cubic, quartic, . . . this touches on the Abel-Ruffini Theorem. Some brilliant and peculiar mathematicians about 200 years ago proved that there are no general solutions/equations for polynomial degree 5 or higher. The degree being the highest integer power of x.

Finding solutions for these polynomials are done algorithmically. Newton's method, which you may have learned about, is one of them.