📈 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.