Linear Equations

# 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\$: