In coordinated geometry on the Cartesian plane
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Gaussian elimination is used to solve systems of linear equations.
Quality does not normally play any part in linear equations.
Equations are not linear when they are quadratic equations which are graphed in the form of a parabola
You would solve them in exactly the same way as you would solve linear equations with real coefficients. Whether you use substitution or elimination for pairs of equations, or matrix algebra for systems of equations depends on your requirements. But the methods remain the same.
They are not. A vertical line is not a function so all linear equations are not functions. And all functions are not linear equations.