Calculate the standard form (Ax + By = C) of a linear equation from a slope and y-intercept, two points, or a slope and a point, with integer coefficients.
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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
To start from two points, first find the slope, then use one point to find b:
m = (y2 - y1) / (x2 - x1)
Where:
- A, B, and C are integers, and A is non-negative 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, which is 1x + 0y = k. A horizontal line is y = k, which is 0x + 1y = k. The calculator handles both cases.
