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Point Slope Form Calculator

Enter a slope and a point or two points to get the equation of a straight line, with a clear step-by-step solution and conversions to other line forms.

I Want To Calculate:

X₁:

Y₁:

Slope (m):

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Point Slope Form Calculator

The Point Slope Form Calculator allows you to quickly determine the equation of a line using a known slope and point or using two points. It provides step-by-step solutions and converts the results into different linear equation forms.

Option 1: Known slope (m) and a point (x₁, y₁)
Option 2: Two points (x₁, y₁) and (x₂, y₂)

The calculator shows how to form the point-slope equation using y - y₁ = m(x - x₁), then converts it into slope-intercept form (y = mx + b) and standard form (Ax + By = C). It handles integers, decimals, fractions, and vertical lines effectively.

How to Use the Point Slope Form Calculator

Mode A: Using Slope and Point

Enter the slope m and the point (x₁, y₁).
Click Calculate.
Calculator outputs:
1. Point-slope equation (step-by-step)
2. Slope-intercept form (y = mx + b)
3. Standard form (Ax + By = C)
4. Graph showing the line passing through the point with slope m

Mode B: Using Two Points

Enter two points (x₁, y₁) and (x₂, y₂).
Click Calculate.
Calculator outputs:
- Slope (m)
- Slope-intercept form (y = mx + b)
- Point-slope form (y - y₁ = m(x - x₁))
- Graph showing both points and the line through them

Understanding Slope

The slope of a line measures its steepness, calculated as the ratio of vertical change (rise) to horizontal change (run) between two points.

Positive slope → line rises left to right
Negative slope → line falls left to right
Zero slope → horizontal line
Undefined slope → vertical line

Point-Slope Form Derivation

Starting from the slope formula:

m = (y - y₁) / (x - x₁)

Where:

m = slope
(x₁, y₁) = known point
(x, y) = any point on the line

Multiply both sides by (x - x₁):

y - y₁ = m(x - x₁)

This is the point-slope form of a line.

Finding the Equation of a Line

1️⃣ Using Slope and a Point

Identify the slope m and point (x₁, y₁).
Substitute into y - y₁ = m(x - x₁).
Simplify to slope-intercept or standard form.

Example: Slope m = 3, point (2, 5)

y - 5 = 3(x - 2)

y - 5 = 3x - 6 → y = 3x - 1

2️⃣ Using Two Points

Calculate slope: m = (y₂ - y₁)/(x₂ - x₁).
Substitute slope and one point into y - y₁ = m(x - x₁).
Simplify to slope-intercept or standard form.

Example: Points (0,1) and (4,5)

Slope: m = (5 - 1)/(4 - 0) = 1

Point-slope: y - 1 = 1(x - 0)

Simplify: y - 1 = x → y = x + 1

Interpretation:

Positive slope → line rises
Negative slope → line falls
Vertical line: slope undefined, x = constant
Horizontal line: slope 0, y = constant

Converting Point-Slope to Slope-Intercept Form

Start with y - y₁ = m(x - x₁)
Distribute: y - y₁ = mx - mx₁
Add y₁: y = mx - mx₁ + y₁
Simplify: y = mx + (y₁ - mx₁)

FAQs

What are the different forms of linear equations?

Standard Form: Ax + By = C
Point-Slope Form: y - y₁ = m(x - x₁)
Slope-Intercept Form: y = mx + b

What is the equation of a straight line?

The equation describes all points on a line:

y = mx + b

m = slope
b = y-intercept

References

Khan Academy: Introduction to Point-Slope Form
StudyPug: Linear Equations Using Point-Slope Form

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