Slope Calculator: equation of a line through two points

This calculator determines the slope of a line passing through two given points and finds the line equation in slope-intercept form (y = mx + b).

Slope and line equation formulas

Given two points (x₁, y₁) and (x₂, y₂):

• Slope: m = (y₂ − y₁) / (x₂ − x₁)
• Equation: y = mx + b, where b = y₁ − m×x₁
• Point-slope form: y − y₁ = m(x − x₁)

The slope indicates the line's steepness: positive means it rises, negative means it falls, zero is horizontal and undefined is vertical.

Example 1: positive slope

Problem: Find the line through (2, 3) and (6, 11).

1. Slope:
   • m = (11 − 3) / (6 − 2) = 8 / 4 = 2.
2. y-intercept:
   • b = 3 − 2×2 = 3 − 4 = −1.
3. Equation:
   • y = 2x − 1.

Answer: m = 2, equation: y = 2x − 1.

Example 2: negative slope

Problem: Find the line through (1, 5) and (4, 2).

1. Slope:
   • m = (2 − 5) / (4 − 1) = −3 / 3 = −1.
2. Intercept:
   • b = 5 − (−1)×1 = 5 + 1 = 6.
3. Equation:
   • y = −x + 6.

Answer: m = −1, equation: y = −x + 6.

Usos comuns

• Determining the incline of ramps, roofs and roads in engineering.
• Analyzing linear trends in economic and statistical data.
• Solving analytic geometry problems in mathematics.
• Computing rates of change in physics and applied sciences.
• Checking whether two lines are parallel (same slope) or perpendicular (negative reciprocal slopes).
• Modeling linear relationships between variables in research.

Common mistakes when calculating slopes

• Reversing the coordinate order when subtracting (using x₁ − x₂ instead of x₂ − x₁).
• Confusing the slope with the y-intercept.
• Not detecting vertical lines where x₂ − x₁ = 0 (undefined slope).
• Mixing up x and y coordinates when substituting into the formula.

Dica profissional

If two lines have the same slope, they are parallel. If the product of their slopes is −1, they are perpendicular. These quick rules let you verify geometric relationships without graphing.