Linear Equations

Definition: An equation in the form $a_0 + a_1x_1 + a_2x_2 + ... + a_nx_n = 0$ is called a Linear Equation of the $n$-variables $x_1, x_2, ..., x_n$.

If we are dealing with equations of two variables, we will often use $x$ and $y$ instead of $x_1$ and $x_2$. For example, the equation $2 + 3x + 4y = 0$ is a linear equation. If we are dealing with equations of three variables, we will often use $x$, $y$ and $z$ instead of $x_1$, $x_2$, and $x_3$. For example, the equation $1 + 3x - 5y + 4z = 0$ is a linear equation. If we are dealing with equations of $n$-many variables, we will often use the former notation. For example, the equation $2 + 3x_1 + x_2 - 4x^3 + 0x^5 = 0$ represents a linear equation of $5$ variables (although the variable $x^4$ is not present in this linear equation).

We will now review what linear equations of 2 variables and 3 variables represent geometrically.

Linear Equations of 2-Variables

A linear equation of 2 variables can be conventionally written in the form $a_1x_1 + a_2x_2 = b$, however, the form $y = mx + b$ where $m$ is the slope of the line formed from the equation and $b$ is the $y$-intercept is commonly used as well. 2-variable linear equations form lines on a 2-dimensional coordinate system (typically the $xy$-coordinate or Cartesian coordinate system). For example, the following graph depicts the equation $x - y = -2$:

Linear Equations of 3-Variables

A linear equation of 3 variables written in the form $a_1x_1 + a_2x_2 + a_3x_3 = b$ or $ax + by + cz = d$ represents a plane in 3-dimensional space, typically an $xyz$-coordinate system. For example, the following graph depicts the linear equation $x + y + 3z = 2$: