There are two eqivalent versions:
y = mx + c
This is called the slope-intercept form. When the graph of the equation is plotted, the line has a slope = m and y-intercept = c.
ax + by + k = 0
This is the standard form and can be generalised to three (or more) dimensions by the simple addition of the term cz (and dw etc) to the left hand side.
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Ax+By=C
Equations are not linear when they are quadratic equations which are graphed in the form of a parabola
They are not. A vertical line is not a function so all linear equations are not functions. And all functions are not linear equations.
Linear equations are a small minority of functions.
Most functions are not like linear equations.