Convert slope-intercept, point-slope or two-point equations to standard form Ax + By = C with integer coefficients, or standard form to slope-intercept.
Related Calculators
- Multiplying Monomials Calculator
- Reciprocal Calculator
- Horner’s Rule Calculator
- Cardano’s Formula Calculator
- All Math and Numbers Calculators
Standard Form Formula
The standard form of a linear equation is written with the variable terms on the left and the constant on the right:
Ax + By = C
To convert from slope-intercept form (y = mx + b), move the x term to the left side:
mx - y = -b
For two distinct points, collect coefficients directly. This also handles vertical lines without dividing by the difference in x-coordinates:
(y2 - y1)x + (x1 - x2)y = (y2 - y1)x1 + (x1 - x2)y1
Where:
- A, B, and C are integers with no common factor. At least one of A and B is nonzero for a line; the first nonzero variable coefficient is positive by convention.
- m is the slope of the line.
- b is the y-intercept, the value of y where the line crosses the y-axis.
- (x1, y1) and (x2, y2) are two points on the line.
After rearranging, multiply through by the smallest number that clears any fractions, then divide A, B, and C by their greatest common factor so the coefficients are the smallest possible integers.
Forms of a Linear Equation
The same line can be written in several equivalent forms. Use this table to see how they relate.
| Form | Equation | Use |
|---|---|---|
| Slope-intercept | y = mx + b | Reading slope and intercept directly |
| Point-slope | y – y1 = m(x – x1) | Writing a line from one point and the slope |
| Standard | Ax + By = C | Integer coefficients, intercept problems |
Sample Conversions
| Slope-intercept | Standard form |
|---|---|
| y = 2x + 3 | 2x – y = -3 |
| y = (1/2)x – 4 | x – 2y = 8 |
| y = -3x + 6 | 3x + y = 6 |
| y = 5 (horizontal) | y = 5 |
Example Problems
Example 1: From slope-intercept form. Convert y = (2/3)x – 4 to standard form. Move the x term to the left to get -(2/3)x + y = -4. Multiply every term by 3 to clear the fraction: -2x + 3y = -12. Multiply by -1 so A is positive: 2x – 3y = 12. The standard form is 2x – 3y = 12.
Example 2: From two points. Find the standard form of the line through (1, 2) and (3, 8). The slope is m = (8 – 2) / (3 – 1) = 3. Using the first point, b = 2 – 3(1) = -1, so y = 3x – 1. Move the x term: 3x – y = 1. The standard form is 3x – y = 1.
FAQ
What is standard form of a linear equation? Standard form is Ax + By = C, where A, B, and C are integers and A is not negative. It keeps both variables on one side and the constant on the other.
Why are the coefficients written as integers? Standard form by convention uses whole-number coefficients with no common factor. If your equation has fractions, multiply every term by the least common denominator, then divide by the greatest common factor of A, B, and C.
Can a vertical or horizontal line be written in standard form? Yes. A vertical line is x = k and a horizontal line is y = k; clear any fractional k to obtain integer coefficients. The calculator handles both cases. Coincident points do not define a unique line. Two-sided coefficients can also simplify to an identity (all points) or a contradiction (no points).
