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

### Example

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

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

• $\text{E: } \frac{x}1+\frac{y}6=1$

$X(1|0|0)$, $Y(0|6|0)$, $Z$ 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.

### Example

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

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