Standard Form (Linear)
Standard form of a linear equation is Ax + By = C, where A, B, and C are integers and A is non-negative.
Formula
Ax + By = C
Definition
Standard form writes a line as $Ax + By = C$, with both variables on the left and the constant on the right; formally, $A$, $B$, and $C$ are integers with $A$ greater than or equal to $0$ and $\gcd(A, B, C) = 1$. Unlike slope-intercept form, which privileges $y$, standard form treats $x$ and $y$ symmetrically. Geometrically it is the normal form of the equation of a hyperplane (line) in $\mathbb{R}^2$, where the vector $(A, B)$ is the normal vector to the line, and a point $(x, y)$ lies on the line if and only if $(A, B) \cdot (x, y) = C$.
Example
For $2x + 3y = 12$, setting $y = 0$ gives the x-intercept $x = 6$, and setting $x = 0$ gives the y-intercept $y = 4$. To convert $y = \frac{2}{3}x - 4$ to standard form, multiply every term by $3$ to get $3y = 2x - 12$, then rearrange and adjust signs: $2x - 3y = 12$. For two lines $2x + 3y = 6$ and $4x - y = 10$, written in matrix form $[[2,3],[4,-1]][[x],[y]] = [[6],[10]]$, the determinant $2(-1) - 3(4) = -14$ confirms a unique solution.
Key Insight
Standard form is handy for finding intercepts quickly, cover one variable to find where the line hits each axis, and it is preferred in systems of equations because it aligns x-coefficients and y-coefficients in neat columns for elimination. That normal-vector view connects standard form to dot products and to solving systems via Cramer's rule or row reduction.