This point slope form calculator writes the equation of a line from one point and a slope — and converts it to slope-intercept and standard form so every representation agrees.

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y − y₁ = m(x − x₁) — the Line’s Native Form

Point-slope is the definition of slope rearranged: rise/run measured from a known anchor. Two-point slope itself is our Slope Calculator; lines from two points in one pass live on the Slope-Intercept page. Part of our Math Calculators collection.

Worked Example (Verified)

  • Point (2, 3), slope 4 (defaults): y − 3 = 4(x − 2) → y = 4x − 8 + 3 = y = 4x − 5 → standard: 4x − y = 5
  • Self-check: at x = 2, 4(2) − 5 = 3 ✔ the anchor lies on its own line (always test this; sign slips in distribution are the #1 point-slope injury)
  • Horizontal case: m = 0 through (2, 3) gives y = 3 — point-slope collapses politely; vertical lines (undefined m) have no point-slope equation at all: x = 2 is the whole story, stated honestly
  • Fraction slope: m = 2/3 ≈ 0.667 through (0, 1): y = 0.667x + 1 — decimals accepted here; exact fraction work prefers clearing denominators into standard form (2x − 3y = −3)

Which Form When

  • Point-slope: building the equation from raw data (a point + a rate), tangent lines in calculus drafts.
  • Slope-intercept: graphing and reading (slope and crossing visible at a glance).
  • Standard form: systems of equations, integer-coefficient requirements, intercepts hunting.
  • All three are one line wearing different typography — convert freely, verify with a point.

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Frequently Asked Questions

What is point-slope form?

y − y₁ = m(x − x₁): the equation of a line built from a known point (x₁, y₁) and slope m. It is literally the slope definition (y − y₁)/(x − x₁) = m with the denominator cleared. Substitute the point and slope, then simplify into whichever form you need.

How do I convert point-slope to slope-intercept?

Distribute m, then isolate y: y − 3 = 4(x − 2) → y − 3 = 4x − 8 → y = 4x − 5. The intercept b = y₁ − m·x₁ directly: 3 − 4×2 = −5. This calculator shows both conversions plus standard form.

Can point-slope use any point?

Yes — every point on the line produces an equivalent equation (they simplify to the same slope-intercept form). Use whichever point you’re given; if given two points, first get the slope from them, then either point finishes the job.

What about vertical and horizontal lines?

Horizontal (m = 0): point-slope gives y = y₁ directly. Vertical: slope is undefined, so no m exists to substitute — the equation is simply x = x₁. Any tool that returns a giant number for a vertical slope is dividing by zero creatively; the honest answer is the x = form.

Where is point-slope form used in real life?

Anywhere a rate and a known state coexist: “temperature rises 2°/hour and it’s 68° now” is point-slope in prose; tangent lines in calculus are point-slope with derivative slopes; linear forecasting models anchor on the latest data point the same way.

Is this point slope form calculator free?

Yes — free, instant, no sign-up.