Linear Equations in Two Variables

A linear equation in two variables is the simplest kind of constraint you can put on two quantities: you take each variable, multiply it by a fixed number (its coefficient), add them up, and set the result equal to some target value. For example, $7x_1 - x_2 = -2$ says: "take seven times the first quantity, subtract the second, and you get negative two."

The word linear is key — it rules out squares ($x_1^2$), products ($x_1 x_2$), reciprocals ($1/x_1$), and all other nonlinear combinations. Only weighted sums are allowed. This restriction might seem severe, but it's exactly what makes linear equations so tractable and geometrically clean.

The general form is $a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b$, where the $a_i$ are the coefficients and $b$ is the right-hand side (RHS). At least one coefficient must be nonzero for the equation to be meaningful.