# Point-Slope Form Calculator

Convert point-slope form to slope-intercept and standard form. Enter a point and slope to see y - y₁ = m(x - x₁), y = mx + b, and Ax + By = C.

## What this calculates

Enter a point and slope to instantly see the line equation in point-slope form, slope-intercept form, and standard form. This calculator also finds the x-intercept and y-intercept.

## Inputs

- **Point X coordinate (x₁)** — The x-coordinate of the known point.
- **Point Y coordinate (y₁)** — The y-coordinate of the known point.
- **Slope (m)** — The slope of the line.

## Outputs

- **Point-Slope Form** — formatted as text — y - y₁ = m(x - x₁)
- **Slope-Intercept Form** — formatted as text — y = mx + b
- **Standard Form** — formatted as text — Ax + By = C
- **Y-Intercept (b)** — Where the line crosses the y-axis.
- **X-Intercept** — Where the line crosses the x-axis.

## Details

Point-slope form is one of three standard ways to write the equation of a line. It is especially useful when you know a point on the line and the slope.

**Point-Slope Form:**

y - y₁ = m(x - x₁)

where (x₁, y₁) is a known point and m is the slope.

**Converting to Slope-Intercept Form (y = mx + b):**

Distribute and solve for y:
y - y₁ = m(x - x₁)
y = mx - mx₁ + y₁
y = mx + b, where b = y₁ - mx₁

For point (3, 7) with slope 2: y - 7 = 2(x - 3), which becomes y = 2x - 6 + 7 = 2x + 1.

**Converting to Standard Form (Ax + By = C):**

Rearrange y = mx + b so that x and y terms are on one side. Multiply through to clear fractions if needed, and make A positive by convention.

From y = 2x + 1: -2x + y = 1, or 2x - y = -1.

**When to use each form:**
- Point-slope: when you know a point and the slope.
- Slope-intercept: when you need to graph quickly or identify the y-intercept.
- Standard: when working with systems of equations or integer coefficients.

## Frequently Asked Questions

**Q: What is point-slope form?**

A: Point-slope form is y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is any point on the line. For a line through (2, 5) with slope 3, the equation is y - 5 = 3(x - 2). This is often the first form you write when given a point and a slope.

**Q: How do I convert point-slope to slope-intercept form?**

A: Distribute the slope and solve for y. From y - 5 = 3(x - 2): y - 5 = 3x - 6, so y = 3x - 6 + 5 = 3x - 1. The slope-intercept form is y = 3x - 1, where the y-intercept is -1.

**Q: What is standard form of a linear equation?**

A: Standard form is Ax + By = C, where A, B, and C are integers and A is positive. It is useful for systems of equations and for quickly finding intercepts. The x-intercept is C/A and the y-intercept is C/B.

**Q: Why are there different forms for the same line?**

A: Each form highlights different properties. Point-slope form is easiest to write from a known point and slope. Slope-intercept shows the y-intercept directly. Standard form works well for systems of equations. All three describe the exact same line.

**Q: How do I find the equation if I have two points instead?**

A: First calculate the slope: m = (y₂ - y₁)/(x₂ - x₁). Then plug m and either point into point-slope form. For points (1, 3) and (4, 9): m = (9-3)/(4-1) = 2, then y - 3 = 2(x - 1), which simplifies to y = 2x + 1.

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Source: https://vastcalc.com/calculators/math/point-slope-form
Category: Math
Last updated: 2026-04-08