Point-Slope Form
Write equation y - y₁ = m(x - x₁) from point and slope. Convert to slope-intercept and standard form. Two points, parallel, perpendicular.
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Why: Understanding point-slope form helps you make better, data-driven decisions.
How: Enter Slope m to calculate results.
Run the calculator when you are ready.
📐 Examples — Click to Load
Point (x₁, y₁) and Slope m
Intercepts & Slope
Slope Magnitude
📐 Calculation Steps
For educational and informational purposes only. Verify with a qualified professional.
📋 Key Takeaways
- • Point-slope form y - y₁ = m(x - x₁) uses one point (x₁,y₁) and slope m — ideal when these are known
- • Slope-intercept form y = mx + b is best for graphing — m is slope, b is y-intercept
- • Standard form Ax + By = C is preferred for systems of equations and integer coefficients
- • Parallel lines have the same slope m
- • Perpendicular lines have slopes that are negative reciprocals: m₂ = -1/m₁
💡 When to Use Each Form
Point-Slope
When you know one point and the slope. Common in calculus for tangent lines.
y - y_{1} = m(x - x_{1})
Slope-Intercept
Best for graphing. Read slope and y-intercept directly.
y = ext{mx} + b
Standard Form
For systems of equations, integer coefficients, and analytical geometry.
ext{Ax} + ext{By} = C
Two-Point Form
When you have two points — compute slope first: m = (y₂-y₁)/(x₂-x₁).
m = \text{Delta} y/\text{Delta} x
🔄 Converting Between Forms
Point-slope → Slope-intercept: Expand y - y₁ = m(x - x₁) to get y = mx + (y₁ - mx₁). So b = y₁ - mx₁.
Slope-intercept → Standard: Rearrange y = mx + b to -mx + y = b. Multiply by LCD for integer coefficients.
Two points → Point-slope: m = (y₂-y₁)/(x₂-x₁). Use either point: y - y₁ = m(x - x₁).
📖 Parallel and Perpendicular Lines
Parallel lines never intersect and have the same slope. If line 1 has slope m, any parallel line has slope m.
Perpendicular lines intersect at 90°. If line 1 has slope m, a perpendicular line has slope -1/m. Example: m=2 → perpendicular slope = -1/2.
Example: Perpendicular to y = 2x + 1 through (3,4)
Original slope m = 2. Perpendicular slope = -1/2. Point-slope: y - 4 = (-1/2)(x - 3). Simplify: y = -x/2 + 11/2.
📊 Quick Reference: Three Forms
| Form | Equation | Use When |
|---|---|---|
| Point-Slope | y - y₁ = m(x - x₁) | Know point and slope |
| Slope-Intercept | y = mx + b | Graphing, read m and b |
| Standard | Ax + By = C | Systems, integer coeffs |
⚠️ Special Cases
- Horizontal line (m=0): y = y₁. Example: y = 5.
- Vertical line: Slope undefined. Use x = x₁. Example: x = 3. Cannot write in slope-intercept form.
- Line through origin: If (0,0) is on the line, b=0, so y = mx.
❓ FAQ
What is point-slope form?
y - y₁ = m(x - x₁), where (x₁,y₁) is a point on the line and m is the slope. Directly expresses the line using one point and slope.
How do I convert to slope-intercept form?
Expand: y - y₁ = mx - mx₁. Add y₁: y = mx + (y₁ - mx₁). So b = y₁ - mx₁.
How do I find the equation from two points?
First find slope: m = (y₂-y₁)/(x₂-x₁). Then use point-slope with either point.
What is the slope of a line perpendicular to y = 2x + 1?
Original slope is 2. Perpendicular slope = -1/2.
When is slope undefined?
For vertical lines when x₁ = x₂. The line has no slope; use x = x₁.
📝 Worked Examples
📐 Real-World: Cost Function
A business has fixed cost $5 and variable cost $2 per item. Total cost C = 5 + 2n where n = number of items. In point-slope form with point (0,5) and slope 2: C - 5 = 2(n - 0). Slope-intercept: C = 2n + 5. The slope represents marginal cost ($2/item).
📌 Summary
Point-slope form y - y₁ = m(x - x₁) is ideal when you know one point and the slope. Convert to slope-intercept for graphing, or standard form for systems. Parallel lines share the same slope; perpendicular lines have slopes that multiply to -1. Use two-point mode when given two points — the calculator computes the slope automatically.
📐 Calculus Connection
In calculus, the point-slope form arises naturally when finding the equation of a tangent line. If f(x) is differentiable at x₁, the tangent line has slope m = f'(x₁) and passes through (x₁, f(x₁)). So the tangent line is y - f(x₁) = f'(x₁)(x - x₁). This is exactly point-slope form.
🎓 Practice Problems (Try in Calculator)
🔗 Next Steps
Explore the Slope Calculator for slope between two points, the Distance Formula Calculator for line length, or the Midpoint Calculator for the midpoint of a segment.
⚠️ Disclaimer: Enter point (x₁,y₁) and slope m for point-slope form. Enable two-point mode to compute slope from (x₁,y₁) and (x₂,y₂). For vertical lines, enter a very large slope (e.g. 999999) — the calculator will show x = x₁.
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