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Solving quadratic systems with elimination

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Solving systems of equations is solving where the equations are equal. Dependency of variables is not the same as dependence of equations themselves. Equality is an equivalence relation meaning that a = b implies a + c = b + c, ra = rb for real r, and if a = b and c = d then a + c = b + d. These properties allow us to manipulate them in such a way that we can eliminate variables while still being true. Testing variables doesn’t always work as they can imply failure of the test.
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You can't replace y with 0 because you'd be missing any solutions where y wasn't zero.    And there may not even be any solutions where y=0.

The system  y=x\^(2)+1  and y=3-x\^(2)   has two solutions.   Neither solution has y=0.

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