Math Planes Intercept equation

Intercept equation of a plane

The intercept equation is a special case of the cartesian equation. It looks like this:

$\text{E: } \frac{x}a+\frac{y}b+\frac{z}c=1$

The special feature is that the intercepts of the plane can be read directly.

  • $a$ is the x-intercept
  • $b$ is the y-intercept
  • $c$ is the z-intercept


  • $\text{E: } \frac{x}4+\frac{y}3+\frac{z}3=1$

    $X(4|0|0)$, $Y(0|3|0)$, $Z(0|0|3)$

  • $\text{E: } \frac{x}1+\frac{y}6=1$

    $X(1|0|0)$, $Y(0|6|0)$, $Z$ does not exist

  • $\text{E: } 2x+4y+z=1$

    $\text{E: } \frac{x}{\frac12}+\frac{y}{\frac14}+\frac{z}1=1$

    $X(\frac12|0|0)$, $Y(0|\frac14|0)$, $Z(0|0|1)$

Cartesian equation → intercept equation

The intercept equation can be formed using the cartesian equation by deviding by the number on the right side of the equation.


$\text{E: } 2x-2y+3z=6\quad|:6$

$\text{E: } \frac{x}3+\frac{y}{-3}+\frac{z}{2}=1$