Linear Equations

An equation is linear if it has the following form:

\[a_0 + a_1x_1 + a_2x_2 + \ldots + a_nx_n = \sum_{i=0}^{n} a_ix_i = 0 \]

So it can have multiple variables, but the variables are only raised to the power of 1 or 0. The following is not a linear equation:

\[x^2 + 2x + 1 = 0 \]

The above equation is in the so called general form, where \(a_1, a_2, \ldots, a_n\) are the coefficients of the variables \(x_1, x_2, \ldots, x_n\) respectively and \(b\) is the constant term. Importantly to the right of the equal sign is 0. In the standard form, the constant term is move to the right side of the equation.

\[a_1x_1 + a_2x_2 + \ldots + a_nx_n = b \]

Visualizing Equations

1 Variable

2 Variables

slope-intercept form: \(y = mx + b\)

n Variables

Solving Equations

System of Linear Equations

\[\begin{vmatrix} a_{11}x_1 + a_{12}x_2 + \ldots + a_{1n}x_n = b_1 \\ a_{21}x_1 + a_{22}x_2 + \ldots + a_{2n}x_n = b_2 \\ \vdots \\ a_{m1}x_1 + a_{m2}x_2 + \ldots + a_{mn}x_n = b_m \end{vmatrix} \]