Standard form of a linear equation
Also known as: standard form, general form of a line
The standard form of a linear equation is Ax + By = C, where A, B, and C are constants (usually integers) and A and B are not both zero. It puts both variables on the same side of the equation.
The standard form of a linear equation is Ax + By = C, with both variable terms on the left and the constant on the right. By convention, A, B, and C are integers with no common factor, and A is non-negative. For example, 2x + 3y = 12 is in standard form; y = −(2/3)x + 4 is the same line written in slope-intercept form.
Standard form makes intercepts easy. Set y = 0 to find the x-intercept (x = C / A) and x = 0 to find the y-intercept (y = C / B). For 2x + 3y = 12, the intercepts are (6, 0) and (0, 4) — two points that let you graph the line immediately. The slope can be read from the coefficients as m = −A / B, which is −2/3 here.
Each form of a line has its specialty: slope-intercept form (y = mx + b) displays slope and y-intercept, point-slope form handles a known point and slope, and standard form handles intercepts cleanly, represents vertical lines (x = 5 fits standard form but not slope-intercept), and suits systems of equations solved by elimination.
Converting between forms is a core algebra skill on standardized tests. The ACT and SAT both ask you to rewrite equations in standard form, extract the slope or intercepts from Ax + By = C, and match equations to graphs — so practice moving fluidly between forms.
Key takeaways
- Standard form is Ax + By = C, with A, B, C typically integers and A ≥ 0.
- Intercepts come quickly: x-intercept at C / A, y-intercept at C / B.
- The slope of Ax + By = C is −A / B.
- Standard form can represent vertical lines, which slope-intercept form cannot.
- The ACT and SAT test converting between standard, slope-intercept, and point-slope forms.