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Line Equation

A line in the plane can be expressed as y = mx + b (slope-intercept), y − y₁ = m(x − x₁) (point-slope), or Ax + By + C = 0 (standard form). Slope m = (y₂−y₁)/(x₂−x₁).

Concept Fundamentals
m = (y₂−y₁)/(x₂−x₁)
Slope
y = mx + b
Slope-Intercept
y − y₁ = m(x − x₁)
Point-Slope
m₁ × m₂ = −1
Perpendicular
Line Equation CalculatorEnter two points, point+slope, or slope+intercept to get all line forms

Why This Mathematical Concept Matters

Why: Line equations appear in statistics (regression), construction (roof pitch), civil engineering (road grade), and coordinate geometry.

How: From two points: compute slope m, then use point-slope or solve for b. Vertical lines: x = constant. Horizontal: y = constant.

  • ●Perpendicular lines have slopes that multiply to −1: m₂ = −1/m₁.
  • ●Roof pitch 6/12 means slope m = 0.5; road grade 6% means m = 0.06.
  • ●Linear regression finds the best-fit line y = mx + b through data points.
Sources:Khan Academy - Linear EquationsWolfram MathWorld - Line
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LINE EQUATION2D Geometry

Line Equation — y = mx + b and More

Two points, point-slope, or slope-intercept. Get all forms, intercepts, and angle.

Slope →Distance →Midpoint →

📐 Examples — Click to Load

Inputs

⚠️For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📐

Slope m = (y₂−y₁)/(x₂−x₁) — rise over run.

— Formula

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Perpendicular lines: m₁ × m₂ = −1.

— Property

📋 Key Takeaways

  • • Slope-intercept y = mx + b: m = slope, b = y-intercept — easiest for graphing
  • • Point-slope y − y₁ = m(x − x₁): use when you know one point and the slope
  • • Standard form Ax + By + C = 0: general form, works for vertical lines (B=0)
  • • Slope m = (y₂−y₁)/(x₂−x₁); angle with x-axis = arctan(m)
  • • Vertical lines (x = constant) have undefined slope; horizontal lines (y = constant) have m = 0

💡 Did You Know?

📐The slope-intercept form y = mx + b is attributed to René Descartes and the development of coordinate geometrySource: History
📊In statistics, linear regression finds the best-fit line y = mx + b through data points — minimizing squared errorsSource: Statistics
🏗️Roof pitch is often given as rise/run — the slope of the roof line. A 6/12 pitch means m = 0.5Source: Construction
🛣️Road grade is slope as percentage: 6% grade = slope 0.06. Critical for drainage and safetySource: Civil Engineering
📈Trend lines in finance use linear equations to project stock prices or economic indicatorsSource: Finance
📏Perpendicular lines have slopes that multiply to −1: m₁ × m₂ = −1. So m₂ = −1/m₁Source: Geometry

📖 Formulas

m=fracy2−y1x2−x1m = \\frac{y_2 - y_1}{x_2 - x_1}

Slope from two points

y=mx+by = mx + b

Slope-intercept

y−y1=m(x−x1)y - y_1 = m(x - x_1)

Point-slope

🎯 Expert Tips

💡 Two Points

Slope m = (y₂−y₁)/(x₂−x₁). Then use point-slope with either point to get y = mx + b.

💡 Perpendicular

Perpendicular slope = −1/m. For y=2x+1, perpendicular through (4,3) has m=−½.

💡 Vertical/Horizontal

Vertical: x = constant (slope undefined). Horizontal: y = constant (m=0).

💡 Intercepts

x-intercept: set y=0, solve for x. y-intercept: set x=0, solve for y (or use b in y=mx+b).

📊 Quick Facts

y=mx+b
Slope-Intercept
3
Input Modes
arctan(m)
Angle
Ax+By+C=0
Standard

❓ FAQ

What is slope-intercept form?

y = mx + b, where m is the slope and b is the y-intercept. The most common form for graphing.

How do I find slope from two points?

m = (y₂ − y₁) / (x₂ − x₁). Rise over run.

What is point-slope form?

y − y₁ = m(x − x₁). Use when you know one point (x₁,y₁) and the slope m.

What about vertical lines?

Vertical lines have undefined slope. Equation is x = constant (e.g., x = 5).

How do I find the x-intercept?

Set y = 0 and solve for x. For y = mx + b, x-intercept = −b/m.

What is the angle with the x-axis?

Angle = arctan(m) in degrees. Slope 1 → 45°, slope 0 → 0°.

How do perpendicular slopes relate?

m₁ × m₂ = −1. So if one line has slope 2, the perpendicular has slope −½.

What is standard form?

Ax + By + C = 0. General form. When B=0, the line is vertical.

⚠️ Disclaimer: Results assume exact arithmetic. For vertical lines, slope-intercept and point-slope forms do not apply.

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