# Symmetric Form of the Equation of a Line

## Symmetric Form of the Equation of a Line

The symmetric form of the equation of a line is an equation that presents the two variables x and y in relationship to the x-intercept a and the y-intercept b of this line represented in a Cartesian plane.

The symmetric form is presented like this: $$\dfrac{x}{a} + \dfrac{y}{b} =1$$, where a and b are non-zero.

It should be noted that the symmetric form of the equation of a line does not make it possible to find the slope of a line. To do that, you must use the relationship $$\small{m = \dfrac{-b}{a}}$$ where $$m$$ represents the slope of the line. This result is obtained by isolating $$y$$ in the equation $$\small{\dfrac{x}{a} + \dfrac{y}{b} =1}$$ multiplying the two sides by $$b$$ to obtain the functional form of this equation: $$\small{y = \dfrac{–b}{a} x + b}$$, in which the coefficient of x is the slope of the line.

### Example

In this illustration, we can see that the coordinates at the origin of the line are ($$\frac{2}{3}$$, 0) and (0, 2), or a = 0.67 and b = 2.

Therefore, the equation of the line in its symmetric form is $$\dfrac{x}{0,67} + \dfrac{y}{2} = 1$$, or $$\dfrac{x}{(\frac{2}{3})} + \dfrac{y}{2} =1$$.

The slope of this line is $$m = \dfrac{-2}{0,67}$$ or $$–3$$.