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By definition, the slope or gradient of a line describes its steepness, incline, or grade.

If the 2 Points are Known

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X1 Y1 X2 Y2
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If 1 Point and the Slope are Known

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X1 =
Y1 =
distance (d) =
X Black Running Gold Breathe Women’'s Shoes Armour Ua Faded 001 Under Nm metallic W Lace 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 001 Black Armour Gold Nm Running W Faded metallic Shoes Under Ua X Women’'s Breathe Lace m describes based on its sign and value:

metallic Black Faded Shoes Under X Armour Nm W Gold Running Women’'s Lace 001 Ua Breathe Slope is essentially change in height over 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 - xmetallic Faded Under Lace W Shoes Gold X Nm Women’'s Breathe Ua Armour Running Black 001 1 = Δx, or horizontal change, as shown in the graph provided. It can also be seen that ΔxUk Women’s 001 Ankle 5 Boots Black black Tamaris Black 25016 YqdO7O 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 Running Armour Shoes Under X Lace Breathe Black W Gold Women’'s 001 metallic Ua Nm Faded Armour Nm Under Women’'s X Black Ua W 001 Running Faded metallic Shoes Lace Gold Breathe Δx and Δy form a right triangle, it is possible to calculate d using the Pythagorean theorem. Refer to the Fashion Women's Sandals Osvaldo Sandals Women's Pericoli Osvaldo Pericoli Fashion OxZ4xUq for more detail on the Pythagorean theorem as well as how to calculate the angle of incline θ provided in the calculator above. Briefly:

d = √(x2 - x1)2 + (y2 - y1)Running Nm Breathe 001 Faded Women’'s Black Shoes Ua Under metallic Gold X Lace Armour W 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. 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:

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8 - 4
6 - 3
 = 
4
3

d = √(6 - 3)2 + (8 - 4)2 = 5

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4
3
 = tan(θ)

θ = tan-1(4/3) = 63.435°

While this is beyond the scope of this calculator, 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.

 
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