Between the Forms

Standard form and slope-intercept form can describe the same line. For $5x+2y = 10$, subtract $5x$ and divide by 2 to get $y = \frac{-5}{2}x+5$. The slope is $\frac{-5}{2}$.

If B is nonzero, rearranging $Ax+By = C$ gives $y = \frac{-A}{B}x+\frac{C}{B}$, so the slope is $\frac{-A}{B}$. For $3x-2y = 8$, the slope is $\frac{3}{2}$. If $B = 0$, the line is vertical and its slope is undefined.