y − y₁ = m(x − x₁)
Point-slope form calculator.
Give it a point and a slope — or two points — and it writes the line in point-slope form, then converts to slope-intercept and standard form with the working shown.
Fractions welcome — 3/4 stays exact through the whole calculation.
Point-Slope Form
y − 4 = 2(x − 3)
Slope-Intercept Form
y = 2x − 2
Standard Form
2x − y = 2
Slope (m)
2
y-intercept
−2
x-intercept
1
Perpendicular Slope
−0.5
Interactive graph. Drag to pan, or scroll to zoom. With the graph focused, use the arrow keys to pan, plus and minus to zoom, and 0 to reset the view.
Step-by-Step Solution
Where the Formula Comes From
Point-slope form is not a separate rule to memorise — it is the slope formula with one multiplication applied. Take any point (x, y) on a line, a known point (x₁, y₁) on the same line, and the definition of slope:
m = (y − y₁) ÷ (x − x₁)
Multiply both sides by (x − x₁) and the denominator clears, leaving y − y₁ = m(x − x₁). That is the whole derivation. Knowing it means you can rebuild the formula in a couple of seconds if it slips your mind in an exam.
Using It
The appeal is that no algebra is required to write the equation down. Given a slope of 2 through the point (3, 4), substitute directly: y − 4 = 2(x − 3). Done. Compare that with slope-intercept form, which needs you to work out the y-intercept first.
Watch the signs when a coordinate is negative. Through (−2, 5) with a slope of 3, the form is y − 5 = 3(x − (−2)), which tidies up to y − 5 = 3(x + 2). The subtraction in the formula is fixed, so a negative x₁ turns into a plus.
Starting From Two Points Instead
If you have two points rather than a point and a slope, work out the slope first and then use either point. Both choices give a correct — and different-looking — equation for the same line. Expanding them proves it: y − 4 = 2(x − 3) and y − 6 = 2(x − 4) both reduce to y = 2x − 2.
That is worth knowing when your answer does not match the one in the back of the book. Before assuming you are wrong, expand both versions to slope-intercept form and compare. They are often the same line written from a different starting point.
Common Questions
What is point-slope form?
Point-slope form writes a line as y − y₁ = m(x − x₁), where m is the slope and (x₁, y₁) is any point the line passes through. It is the fastest form to write down, because you can drop the two things you know straight into it without solving anything.
When should I use point-slope form instead of y = mx + b?
Use point-slope when you are given a point and a slope, which is the usual starting position in a calculus or physics problem. Slope-intercept form requires you to know where the line crosses the y-axis, and working that out is an extra step you often do not need.
Why is it y − y₁ and not y + y₁?
The form comes straight from the slope formula. Starting from m = (y − y₁)/(x − x₁) and multiplying both sides by (x − x₁) gives y − y₁ = m(x − x₁). The subtraction is what was already there in the slope formula.
Does it matter which point I use?
No. Any point on the line gives a valid point-slope equation, and they all describe the same line. Two people can write the same line as y − 4 = 2(x − 3) and y − 6 = 2(x − 4), and both are correct — expanding either one gives y = 2x − 2.
How do I convert point-slope form to slope-intercept form?
Expand the bracket and move the constant across. From y − 4 = 2(x − 3): multiply out to get y − 4 = 2x − 6, then add 4 to both sides for y = 2x − 2. This calculator shows the same conversion for whatever you enter.