Linear Equation

A linear equation has variables raised only to the first power and no products of variables. In one variable, $ax + b = 0$ has the unique solution $x = -b/a$ (provided $a \neq 0$).

In two variables, $ax + by = c$ describes a straight line in the plane. Common forms:

• Slope-intercept: $y = mx + b$ — easy to graph (slope $m$, y-intercept $b$).
• Point-slope: $y - y_1 = m(x - x_1)$ — easy from a single point.
• Standard: $ax + by = c$ — symmetric, generalises to higher dimensions.

Systems of linear equations are solved by substitution, elimination, or matrix methods (Cramer's rule, Gaussian elimination). Linear equations are the foundation of linear algebra and the simplest models in physics, economics, and machine learning.