The equation of a line is not always given in the form [[$ y = kx + b $]]. When it is important that all the coefficients in the equation are integers, a general form of the line equation is used. In it, all the terms are shifted to the left side of the equation and there is only the number zero on the right side of the equation. It is now easy to multiply both sides of the equation in order to remove fractions, because the right side of the equation will remain zero even if it is multiplied. In addition, the general form is usually written so that the coefficient of [[$ x $]] becomes positive.

Example 1

Determine the slope of the line [[$ x - 2y + 4 = 0 $]].

The equation of the line is now given in a general form from which the slope is not directly visible. Let's transform the equation into its slope-intercept form [[$ y = kx + b $]] by solving the equation for the variable [[$ y $]].

[[$ \quad \begin{align} x - 2y + 4 &= 0 \ \\ - 2y &= -x - 4 \space ||:(-2) \ \\ y &= \dfrac{-x}{-2} - \dfrac{4}{-2} \ \\ y &= \dfrac{1}{2} x + 2 \end{align} $]]​ ​

The terms that do not contain y are moved to the right side of the equation. The coefficient of y is removed by dividing both of sides of the equations by -2.

Answer: The slope of the line is [[$ \displaystyle\frac {1} {2} $]]​.

Example 2

Convert the equation of a line from its slope-intercept form [[$ y = \displaystyle\frac {2} {3}x \: – 4 $]]​ to its general form.

[[$ \quad \begin{align} y &= \dfrac{2}{3} x - 4 \ \\ - \dfrac{2}{3} x + y + 4 &= 0 \space ||·(-3) \ \\ 2x - 3y -12 &= 0 \end{align} $]]​

All terms are moved to the left side of the equation. By multiplying both sides of the equation by -3, we get rid of fractions and the coefficient of x becomes positive.

Answer: The general form of the equation is [[$ 2x – 3y –12 = 0 $]]​.