Use our free online slope calculator to instantly find the slope between two points. Enter coordinates (x₁, y₁) and (x₂, y₂) to calculate slope with step-by-step explanation. Perfect for students, teachers, and professionals.
Enter coordinates to see slope, angle, and distance.
Slope is a fundamental concept in mathematics that measures how steep a line is. In simple terms, slope represents the rate of change between two points on a line. It tells you how much the line rises (or falls) for every unit it moves horizontally.
The slope of a line is its vertical change divided by its horizontal change, commonly expressed as rise over run. When you have two points on a line, the slope is calculated as the change in y divided by the change in x. The slope of a line is a measure of how steep it is, and it is often denoted by the letter m.
Understanding slope is essential in many areas, including algebra, geometry, physics, engineering, and economics. It helps you understand relationships between variables, analyze trends, and solve real-world problems.
To find the slope between two points (x₁, y₁) and (x₂, y₂), we use the slope formula:
m = (y₂ - y₁) / (x₂ - x₁)
Slope = Rise / Run = ΔY / ΔX
To find the slope between two points, follow these three simple steps:
Find the difference between the y coordinates. This is called the "rise" or Δy (change in y).
Find the difference between the x coordinates. This is called the "run" or Δx (change in x).
Divide the change in y by the change in x to get the slope.
⚠️ Important: If Δx = 0 (x₂ = x₁), the line is vertical and the slope is undefined.
Calculate the slope between two points: (2, 5) and (9, 19).
Answer: The slope is 2.
This means the line rises 2 units for every 1 unit it moves to the right.
Calculate the slope between points: (3, 8) and (7, 2).
Answer: The slope is -1.5.
A negative slope means the line falls as it moves to the right.
Calculate the slope between points: (1, 1) and (5, 3).
Answer: The slope is 0.5 (or 1/2).
The line rises 1 unit for every 2 units it moves horizontally.
Once you have the slope m, you can describe the line using different equations. There are three common ways to write line equations with slope:
y = mx + b
This is the most common form. m is the slope, and b is the y-intercept (where the line crosses the vertical y-axis).
Example: y = 2x + 1
y - y₁ = m(x - x₁)
Useful when you know the slope and one specific point (x₁, y₁) on the line, but not necessarily the y-intercept.
Example: y - 5 = 2(x - 2)
Ax + By = C
The equation should not include fractions or decimals, and the x coefficient should only be positive. Useful for systems of equations.
Example: 2x - y = -1
You can convert between these forms. For example, starting with point-slope form y - 5 = 2(x - 2):
If you have the equation for a line, you can put it into slope-intercept form. The coefficient of x will be the slope.
You have the equation of a line: 6x - 2y = 12, and you need to find the slope.
Your goal is to get the equation into slope-intercept format y = mx + b:
The y-intercept of a line is the value of y when x = 0. It is the point where the line crosses the y-axis.
Using the equation y = 3x - 6, set x = 0 to find the y-intercept:
The y-intercept is -6, or the point (0, -6)
The x-intercept of a line is the value of x when y = 0. It is the point where the line crosses the x-axis.
Using the equation y = 3x - 6, set y = 0 to find the x-intercept:
The x-intercept is 2, or the point (2, 0)
If you know the slope of a line, any line parallel to it will have the same slope and these lines will never intersect.
Example: If a line has slope 3, all parallel lines also have slope 3.
If you know the slope of a line, any line perpendicular to it will have a slope equal to the negative inverse of the known slope. Perpendicular means the lines form a 90° angle when they intersect.
Example: If a line has slope -4:
Calculating a line's properties often involves more than just the slope.
Finds the length of the line segment between two points.
d = √[(x₂ - x₁)² + (y₂ - y₁)²]Finds the angle (θ) the line makes with the positive x-axis.
θ = tan⁻¹(m)When you look at a graph, you can visually determine the slope of a line. Understanding how slope appears on a graph helps you interpret data and understand relationships between variables.
To find the slope from a graph visually:
Tip: If the line goes up from left to right, the slope is positive. If it goes down, the slope is negative. The steeper the line, the larger the absolute value of the slope.
In construction (like roads or plumbing), slope is often expressed as a percentage grade. You can convert standard slope to grade using this formula:
A: Slope represents how steep a line is, calculated as the change in y divided by the change in x. It measures the rate of change between two points on a line and is denoted by the letter m. Slope tells you how much the line rises (or falls) for every unit it moves horizontally.
A: To calculate slope between two points (x₁, y₁) and (x₂, y₂), use the formula: m = (y₂ - y₁) / (x₂ - x₁). First, subtract the y-coordinates to find the rise (Δy). Then subtract the x-coordinates to find the run (Δx). Finally, divide rise by run to get the slope.
A: Slope represents the steepness and direction of a line. A positive slope means the line rises from left to right (upward trend). A negative slope means the line falls from left to right (downward trend). Zero slope means the line is horizontal (no change). An undefined slope means the line is vertical (infinite steepness).
A: Yes, slope can be negative. A negative slope indicates that as x increases, y decreases. The line falls from left to right. This is common in scenarios like declining sales, decreasing temperatures, or inverse relationships between variables.
A: The slope formula is m = (y₂ - y₁) / (x₂ - x₁), where m is the slope, (x₁, y₁) and (x₂, y₂) are two points on the line. This formula calculates the ratio of vertical change (rise) to horizontal change (run).
A: If the equation is in slope-intercept form (y = mx + b), the slope is the coefficient of x (the number multiplied by x). For example, in y = 3x + 5, the slope is 3. If the equation is in another form, convert it to slope-intercept form first.
A: Rise over run is another way to describe slope. "Rise" refers to the vertical change (change in y), and "run" refers to the horizontal change (change in x). Slope = rise / run = Δy / Δx. This visual description helps understand slope as how much a line goes up or down for each unit it moves horizontally.
A: Yes, slope can be zero. A zero slope means the line is perfectly horizontal (parallel to the x-axis). This occurs when y₂ = y₁, meaning there is no vertical change. The line has a constant y-value regardless of the x-value.
A: An undefined slope occurs when the line is perfectly vertical (parallel to the y-axis). This happens when x₂ = x₁, meaning there is no horizontal change (run = 0). Since you cannot divide by zero, the slope is undefined. Vertical lines have equations of the form x = constant.