Standard Form (Linear)
AlgebraStandard 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 of a line is written as $Ax + By = C$. Both variables are on the left side, and the number is on the right. $A$, $B$, and $C$ are whole numbers.
Example
$2x + 3y = 12$ is in standard form. To find the x-intercept, set $y = 0$: $2x = 12$, $x = 6$. To find the y-intercept, set $x = 0$: $3y = 12$, $y = 4$.
Key Insight
Standard form is handy for finding intercepts quickly: cover one variable to find where the line hits each axis.
Definition
Standard form $Ax + By = C$ requires $A$, $B$, $C$ to be integers with $A$ greater than or equal to $0$ and $\gcd(A, B, C) = 1$. Any linear equation can be converted to standard form. It treats $x$ and $y$ symmetrically, unlike slope-intercept form, which privileges $y$.
Example
Convert $y = \frac{2}{3}x - 4$ to standard form: multiply every term by $3$ to get $3y = 2x - 12$, then rearrange: $-2x + 3y = -12$, and multiply by $-1$: $2x - 3y = 12$.
Key Insight
Standard form is preferred in systems of equations because it aligns x-coefficients and y-coefficients in neat columns, making the elimination method straightforward.
Definition
Standard form $Ax + By = C$ is the normal form of the equation of a hyperplane (line) in $\mathbb{R}^2$. The vector $(A, B)$ is the normal vector to the line. For two non-parallel lines $A_1x + B_1y = C_1$ and $A_2x + B_2y = C_2$, the system can be written as the matrix equation $[A_1, B_1; A_2, B_2][x; y] = [C_1; C_2]$, solved by Cramer's rule or row reduction.
Example
Lines $2x + 3y = 6$ and $4x - y = 10$ in matrix form: $[[2,3],[4,-1]][[x],[y]] = [[6],[10]]$. The determinant is $2(-1) - 3(4) = -14$, confirming a unique solution.
Key Insight
The normal vector $(A, B)$ is perpendicular to the line, a fact that connects standard form to dot products: a point $(x, y)$ lies on the line iff $(A, B) \cdot (x, y) = C$.