Slope of line

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The slope of any given line is the steepness alongside the path of any given line. Without the help of a compass, one can determine if any two of the supplied lines lie parallel to one another, perpendicular, or not by calculating the slope of the lines.

 

The slope of almost any given line may be evaluated utilizing any 2 separate places or two separate points on the given line. The formula to calculate the slope of a line is known to evaluate the ratio of change in vertical points to the change in the horizontal points of a given line.

 

Definition of Slope

We can define the slope of any given line as the ratio of change in the y axis(difference of the points in the y-axis) to the ratio of change in the x-axis (difference of the points in the x coordinate). The change in the y-axis is denoted as  Δy, whereas the rate for the x-axis is denoted as  Δx. Therefore, the slope(m) is denoted as,

m = Δy/Δx

where m is the gradient of the line

We also know that tan θ = Δy/Δx

Thus we also say ‘tan θ’  to be the slope of the line.

Slope Between Two Points

The slope for any given line can be found utilizing 2 coordinates on the given straight line. If we know the points (x,y) of these two points, we can use this formula ( formula to calculate the slope of the line).

For example we can assume the coordinates to be,

P1 = (x1, y1)

P2 = (x2, y2)

As discussed previously, the slope is defined as the ” ratio of change in y-axis with change in the x-axis of the given line”. Hence, substituting the  given values of Δy and Δx in the equation, we already know that:

Δy = y2 – y1

Δx = x2 – x1

Slope = m = (y2 – y1)/(x2 – x1)

Types of Slope

The slope of the line can be classified depending upon the two variables x and y, and the slope obtained from these variables.

There are 4 different types of slopes, such as

  • Positive slope
  • Negative slope
  • Zero slope
  • Undefined Slope

Positive Slope

While moving left to right in a 2D plane (coordinate plane), the line rises, which also depicts that x and y are directly proportional, or when x increases, y also increases. This is the concept of a positive slope.

Negative Slope

While moving left to right in a 2D plane (coordinate plane), the line falls, which also depicts that x and y are inversely proportional, or when x increases, y decreases. This is the concept of a negative slope.

Zero Slope

The zero slope includes a line that is flat horizontally or which is parallel to the x-axis. This slope will not have any x variable in it if seen in any equation.

Undefined Slope

The slope which is infinitely large or which is parallel to a vertical line ( or simply the slope of a vertical line), is called an undefined slope.

 

The Slope of Parallel Lines

Since the angle of inclination between two parallel lines is equal. Hence, the slopes for the lines are equal.

 

The slope of Perpendicular Lines

Since perpendicular lines have an angle of 90º  between them. Hence, the product of the lines is equal to -1.

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