Ax+By=C
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Equations are not linear when they are quadratic equations which are graphed in the form of a parabola
It does not matter because they are equivalent. You can always convert from a slope-intercept form to a standard linear form (and vice versa).
There are many different standard forms: standard forms of numbers, of linear equations, of circles, etc. The standard form of numbers simplifies working with very large and very small numbers.
If the slopes are the same on both graphs, they are parallel, and will never touch.
They are the simplest form of relationship between two variables. Non-linear equations are often converted - by transforming variables - to linear equations.