Equivalent equation forms
Step-by-step derivation
Verification and analysis
Coordinate graph
Drag to pan. Use the controls to zoom, reset, or download the graph.
Relationship, intersection, and point checks
Coordinate table
Generate sample points from the current equation. These values also appear in exported CSV files.
| Point | x | y | Verification |
|---|---|---|---|
| Generate an equation first. | |||
Straight-line formulas
| Purpose | Formula |
|---|---|
| Slope from two points | m = (y₂ − y₁) / (x₂ − x₁) |
| Slope-intercept form | y = mx + b |
| Point-slope form | y − y₁ = m(x − x₁) |
| Standard form | Ax + By = C |
| Intercept form | x/a + y/b = 1 |
| Angle and slope | m = tan(θ) |
| Perpendicular slope | m₂ = −1/m₁ |
| Point-to-line distance | d = |Ax₀ + By₀ − C| / √(A² + B²) |
Generate a line equation
- Select the calculation mode matching your known values.
- Enter integers, decimals, fractions, or mixed numbers.
- Choose precision, output form, variables, and graph limits.
- Press the generate button to calculate every line form.
- Review the derivation, verification, graph, and coordinate table.
- Copy, print, or export the completed solution.
Example data
| Known information | Generated equation | Notes |
|---|---|---|
| Points (2, 3) and (6, 11) | y = 2x − 1 | Slope equals 2. |
| Slope 3 and point (1, 4) | y = 3x + 1 | Point-slope conversion. |
| Intercepts 4 and 2 | x + 2y = 4 | Both intercepts are finite. |
| Reference y = 2x − 1, perpendicular through (1, 4) | y = −0.5x + 4.5 | Negative reciprocal slope. |
Copy-ready output
Generate an equation to create the complete solution text.
Line equation questions
What information can generate a line equation?
Two points can define one unique non-degenerate line. A point and slope also work. Intercepts or coefficients provide other valid starting forms.
How does the calculator handle a vertical line?
A vertical line has undefined slope. Its equation uses the form x equals a constant. The domain contains one value while the range is unrestricted.
Can I enter fractions or mixed numbers?
Yes, the input parser accepts common fractions. Mixed values such as 2 1/3 are supported. Decimal results can also appear beside fractions.
How is a perpendicular line generated?
Non-vertical slopes use the negative reciprocal rule. Horizontal references produce vertical perpendicular lines. Vertical references produce horizontal perpendicular lines.
What does normalising coefficients mean?
Normalisation removes common integer factors when practical. It also controls the leading coefficient sign. Equivalent equations therefore look simpler and more consistent.
How does best-fit regression work?
Least squares minimises squared vertical residuals. The calculator reports slope, intercept, and R-squared. Duplicate points remain valid unless all x-values match.
Can the equation be converted between forms?
Yes, one internal standard representation drives every conversion. Some forms are unavailable for special lines. The result explains each unavailable form clearly.
How is a point checked against a line?
The point is substituted into the standard equation. A near-zero residual means the point lies on it. The calculator also reports perpendicular distance.
Can I save the graph and results?
The graph downloads as a PNG image. Results can export as CSV or PDF. Printing supports paper or browser PDF output.