By definition, the slope or gradient of a line describes its steepness, incline, or grade. m = | y2 - y1 |

x2 - x1 |

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## If the 2 Points are Known

X1 | Y1 | X2 | Y2 | |

## If 1 Point and the Slope are Known

X1 = | |

Y1 = | |

distance (d) = | |

slope (m) = | OR angle of incline (θ) =° |

**Slope, sometimes referred to as gradient in mathematics, is a number that measures the steepness and direction of a line, or a section of a line connecting two points, and is usually denoted by m**. Generally, a line"s steepness is measured by the absolute value of its slope, **m**. The larger the value is, the steeper the line. Given **m**, it is possible to determine the direction of the line that **m** describes based on its sign and value:

Slope is essentially the change in height over the change in horizontal distance, and is often referred to as "rise over run." It has applications in gradients in geography as well as civil engineering, such as the building of roads. In the case of a road, the "rise" is the change in altitude, while the "run" is the difference in distance between two fixed points, as long as the distance for the measurement is not large enough that the earth"s curvature should be considered as a factor. The slope is represented mathematically as:

m = | y2 - y1 |

x2 - x1 |

In the equation above, **y2 - y1 = Δy**, or vertical change, while **x2 - x1 = Δx**, or horizontal change, as shown in the graph provided. It can also be seen that **Δx** and **Δy** are line segments that form a right triangle with hypotenuse **d**, with **d** being the distance between the points **(x1, y1)** and **(x2, y2)**. Since **Δx** and **Δy** form a right triangle, it is possible to calculate **d** using the Pythagorean theorem. Refer to the Triangle chungcutuhiepplaza.com for more detail on the Pythagorean theorem as well as how to calculate the angle of incline **θ** provided in the chungcutuhiepplaza.com above. Briefly:

d = √(x2 - x1)2 + (y2 - y1)2

The above equation is the Pythagorean theorem at its root, where the hypotenuse **d** has already been solved for, and the other two sides of the triangle are determined by subtracting the two **x** and **y** values given by two points.

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Given two points, it is possible to find **θ** using the following equation:

m = tan(θ)

Given the points (3,4) and (6,8) find the slope of the line, the distance between the two points, and the angle of incline:

m = | 8 - 4 |

6 - 3 |

3 |

d = √(6 - 3)2 + (8 - 4)2 = 5

4 |

3 |

θ = tan-1( | 4 |

3 |

While this is beyond the scope of this chungcutuhiepplaza.com, aside from its basic linear use, the concept of a slope is important in differential calculus. For non-linear functions, the rate of change of a curve varies, and the derivative of a function at a given point is the rate of change of the function, represented by the slope of the line tangent to the curve at that point.