Mathematics ยท Algebra
Forms of Linear Equations
Reference entry · last updated September 20, 2026
A linear equation in two variables is an equation whose graph in the coordinate plane is a straight line. The slope \(m\) is the constant rate of change of \(y\) with respect to \(x\), and every form of a linear equation encodes the same line with a different emphasis.
1. First principles: slope
For two distinct points \((x_1, y_1)\) and \((x_2, y_2)\) with \(x_1 \neq x_2\), the slope is the ratio of the vertical change to the horizontal change:
For a straight line this ratio is the same for every pair of distinct points, which is what makes the line straight. A positive slope rises left to right, a negative slope falls, and a zero slope is horizontal [1].
2. Slope-intercept form
Here \(m\) is the slope and \(b\) is the \(y\)-intercept, the value of \(y\) where the line crosses the \(y\)-axis at \(x = 0\). This form is the fastest way to graph a line: plot \((0, b)\), then move according to the slope.
3. Point-slope form
The form follows directly from the slope definition. Rearranging \( m = (y - y_1)/(x - x_1) \) gives it. Use it when a point and the slope are known but the \(y\)-intercept is not [1, 2].
4. Standard form
Common conventions require \(A\), \(B\), and \(C\) to be integers, \(A \ge 0\), and \(\gcd(|A|, |B|, |C|) = 1\). The form treats \(x\) and \(y\) symmetrically and is convenient for solving systems. When \(B \neq 0\) the slope is \(m = -A/B\).
5. Converting between forms
| Form | Equation | Best for |
|---|---|---|
| Slope-intercept | \( y = mx + b \) | Graphing; reading slope and intercept |
| Point-slope | \( y - y_1 = m(x - x_1) \) | A known point and slope |
| Standard | \( Ax + By = C \) | Systems; integer coefficients |
Any form converts to any other by distribution and rearrangement. To find the slope from standard form, solve for \(y\). To find the \(y\)-intercept from standard form, set \(x = 0\).
6. Special cases and parallel lines
- Vertical line: \( x = k \). The slope is undefined because \(\Delta x = 0\). The line is not a function of \(x\).
- Horizontal line: \( y = b \). The slope is \(0\).
- Parallel lines: equal slopes, \( m_1 = m_2 \), with different intercepts.
- Perpendicular lines: slopes are negative reciprocals, \( m_1 m_2 = -1 \), provided neither is vertical.
See also
- Systems of Linear Equations: Elimination and Substitution
- Linear Systems Classification
- Domain and Range of Real Functions
- Transformations of Functions
References
- ^OpenStax, College Algebra 2e, Rice University (Linear equations in one and two variables).
- ^James Stewart, Lothar Redlin, and Saleem Watson, Precalculus: Mathematics for Calculus, 7th ed., Cengage Learning, 2015.