📈 Slope & Line Equation Calculator
Through (1, 2) and (4, 4): slope m = 2/3 exactly, y-intercept b = 4/3, and the line in all three forms — slope-intercept, point-slope and integer standard form.
Coordinates accept integers, decimals and fractions (1/2).
m = 2/3
slope (exact)
b = 4/3
y-intercept
| Slope-intercept form | y = (2/3)x + 4/3 |
|---|---|
| Point-slope form | y − 2 = (2/3)(x − 1) |
| Standard form | 2x − 3y = -4 |
Working
- Slope = rise ÷ run = (y₂ − y₁) / (x₂ − x₁) = (4 − 2) / (4 − 1) = 2 / 3 = 2/3
- Intercept from y₁ = m·x₁ + b: b = 2 − (2/3)·(1) = 4/3
- Standard form clears the fractions and reduces by the common factor.
Exact fractions throughout — the slope through (1, 2) and (4, 4) is 2/3, not 0.6667. Runs locally.
How the slope & line equation calculator works
From two points the slope is rise over run — (y₂ − y₁)/(x₂ − x₁) — computed as an exact fraction, never a rounded decimal. The y-intercept follows from substituting either point into y = mx + b, again exactly. The tool then writes the line three ways: slope-intercept form (y = mx + b, the graphing form), point-slope form (y − y₁ = m(x − x₁), the form proofs and calculus use), and standard form (Ax + By = C with the fractions cleared to smallest integers). Vertical lines are handled honestly: equal x-coordinates mean an undefined slope, and the tool says so and gives the x = c equation instead of dividing by zero.
The exact fraction matters more here than almost anywhere: a slope of 2/3 written as 0.6667 fails the "is this line through these points?" check by a hair, and standard-form conversion from a rounded decimal produces the wrong integers. Keeping m and b as fractions until the last step is exactly what a maths teacher does on the board — and what decimal-only calculators can't.
Frequently asked questions
How do I find the slope between two points?
Divide the change in y by the change in x: m = (y₂ − y₁)/(x₂ − x₁). Through (1, 2) and (4, 4): m = (4 − 2)/(4 − 1) = 2/3 — kept as a fraction, since 0.6667 is already wrong in the fourth decimal.
What are the three forms of a line equation?
Slope-intercept y = mx + b (best for graphing), point-slope y − y₁ = m(x − x₁) (best when you know a point), and standard Ax + By = C with integer A, B, C (the textbook "final answer" form). The tool derives all three exactly.
What if the two points have the same x?
Then the run is zero and the slope is undefined — the line is vertical, with equation x = c. The tool detects this and says so instead of producing a division-by-zero error or a giant fake slope.
How is standard form derived?
Start from y = mx + b, multiply through by the least common denominator to clear fractions, move x and y to one side, and divide by any common factor — the tool shows the result with A positive by convention.
Can coordinates be fractions or decimals?
Yes — every coordinate is parsed as an exact rational, so points like (1/2, 3/4) or (2.5, −1.25) give exact slopes and intercepts.