The Slope Calculator finds the slope of a line given two points. It also provides the equation of the line in slope-intercept form (y = mx + b) and point-slope form, along with the distance between the two points.
Enter the coordinates of two points (x₁, y₁) and (x₂, y₂) to calculate the slope and line equation instantly.
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The slope of a line measures its steepness and direction. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. A positive slope means the line goes upward from left to right, a negative slope means it goes downward, a slope of zero produces a horizontal line, and an undefined slope (division by zero) produces a vertical line.
The slope-intercept form y = mx + b is the most common way to express a linear equation, where m is the slope and b is the y-intercept. The point-slope form y - y1 = m(x - x1) is useful when you know one point and the slope. Both forms describe the same line and can be converted between each other algebraically.
Parallel lines have identical slopes but different y-intercepts. Perpendicular lines have slopes that are negative reciprocals of each other, meaning their product equals -1. These relationships are fundamental in coordinate geometry and are used extensively in engineering, physics, and computer graphics.
Problema: Find the slope of the line through (1, 2) and (4, 8).
Solucion: m = (8 - 2) / (4 - 1) = 6 / 3
Respuesta: The slope is 2. The equation is y = 2x + 0, or simply y = 2x.
Problema: A line passes through (3, 5) with a slope of -2. Find its equation.
Solucion: Point-slope form: y - 5 = -2(x - 3). Expanding: y - 5 = -2x + 6. Solving: y = -2x + 11.
Respuesta: The equation is y = -2x + 11, with a y-intercept of 11.
Problema: Given m = 3 and b = -6, find the x-intercept.
Solucion: Set y = 0: 0 = 3x - 6. Solving: 3x = 6, so x = 2.
Respuesta: The x-intercept is at (2, 0). The equation is y = 3x - 6.
Problema: Find lines parallel and perpendicular to y = 4x + 1 that pass through (2, 3).
Solucion: Parallel: same slope m = 4. Using 3 = 4(2) + b gives b = -5, so y = 4x - 5. Perpendicular: m = -1/4. Using 3 = -1/4(2) + b gives b = 3.5, so y = -0.25x + 3.5.
Respuesta: The parallel line is y = 4x - 5. The perpendicular line is y = -0.25x + 3.5.
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