Slope Formula: Equation of Line & Derivation

Slope formula helps to calculate the slope of a line. The ratio of change in the X and Y axis is found through the slope formula. These coordinate values have a vital role in geometry. The letter ‘m’ represents the slope. The slope formula is used in several fields including economics, geoscience, and accounting, finance.

The Formula of Slope is:

• Simply, the Slope formula helps us to determine the steepness or inclination of a given line.
• The x and y coordinates of points which lie on the line are used to evaluate the slope of a line.
• To predict whether a line is parallel or perpendicular or at any angle, the slope is measured.

Read more: Angle between Two Lines

Key Terms: Slope of a Line, Straight Lines, Positive Slope, Negative Slope, Equation of Line Graph, X-coordinate, Y-coordinate

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What is a Slope?

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A slope of a line, in maths, can be defined as the change in the y-coordinate with respect to the change in x-coordinate. The net change in the y-coordinate can be denoted by Δy. And the net change in the x-coordinate can be denoted by Δx.

Thus, the change in the y-coordinate with respect to the change in the x-coordinate can be represented by:

⇒ m = change in y/change in x = Δy/Δx

Where “m” is the slope of a line.

What is Slope?

The slope of the line can also be denoted by:

Thus, tan θ is the slope of a line.

Typically, the slope of a line helps to measure the steepness and direction. The change in the “y” coordinate with respect to the “x” coordinates is referred to as the slope of a line. It is denoted by the letter “m”. 

Read More:

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Slope Formula

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The slope of a line is calculated using the X and Y-axis. Consider two points of a line as F (x₁,y₁) and G(x₂,y₂). Hence, the formula of slope becomes,

⇒ Slope m = Change in Y-axis/ Change in X-axis

Thus,

• In other words, the slope of a line increases as it passes from the Y to the X-axis. So,
• Slope m = Y axis/X axis

• The slope of a line between two points is also referred to as the rise of the line from one point to another point (along the y-axis) over the run (along the x-axis). Therefore, Slope, m = Rise/Run.

Also Read: Height and Distance 

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Calculating Slope of a Line

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Consider two points of a line as (x₁, y₁) and (x₂, y₂). Three steps are essential to obtain a straight-line slope. 

• Ensure the two points are in a line.
• Choose one side as (x₁, y₁) and another as (x₂, y₂).
• Find the slope using the slope line formula. 

Several facts are vital in finding the slope of a line:

• The slope formula results in positive or negative.
• A rise in the line occurs during a positive value in slope.
• The absence of slope is seen in vertical lines
• Equal slope can be found in parallel lines. On another side, horizontal lines form a zero slope.

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Derivation of Slope Formula

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The x and y coordinates of the line are used to calculate the slope of the line. The net change in y-coordinate is Δy, while the net change in the x coordinate is Δx. So the change in y-coordinate with respect to the change in x-coordinate can be written as,

where

• m = slope
• Δy = change in y-coordinates
• Δx = change in the x-coordinates

We are aware that tan θ can also be known as the slope of the line wherein θ is the angle made by the line with the x-axis’ positive direction.

Thus, tanθ = height/base

Now, the height/base between any two points = (y₂ - y₁)/(x₂ - x₁)

Thus, the slope equation is going to be, m = tanθ = Δy/Δx

From a graph, we can observe:

• Δy = (y₂ - y₁)
• Δx = (x₂ - x₁)

Thus, the slope formula can be represented by: Slope = m = (y₂ - y₁)/(x₂ - x₁)

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Slope of a Line

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The range of slope varies in different lines. Let us look at the formation of decline in various lines. Thus, in the equation of a line formula, the following are used:

Perpendicular Lines

Consider two perpendicular lines, I₁ and I₂. By using the properties of angles, β = α + 90°. From the slopes, we get that the product of perpendicular lines of a slant is equal to -1. 

Vertical Lines

There is no slope in vertical lines. The rising and falling of the angle are not determined in vertical lines. Value is absent in x coordinates. According to slope formula of line, m = (y₂ – y₁)/(x₂ – x₁). In the case of vertical lines, x₂ = x₁ = 0. Therefore, the value of slope in y-coordinates is zero. 

What is a Slope Infograph

Also Read: 

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Difference between Positive and Negative Slope

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There are positive and negative slopes. The table given below depicts the difference between these two slopes.

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Slope of Line Equation

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The slope of a line equation is:

where,

• m = slope
• Δy = change in y-coordinates
• Δx = change in the x-coordinates
• θ = angle made by the line with the positive x-axis

The equation for the point-slope of a line given as,

y − y₁ = m(x − x₁)

Therefore, the slope-intercept is,

y = mx + b. 

Here, b is y-intercept. 

Also Read: Distance between Two Points

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Things to Remember

• The slope of a line can be determined through the slope formula. The ratio of change in the X and Y axis is found through the slope formula. 
• Th formula of slope is given by: \(\begin{array}{l}m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\end{array}\).
• The product of perpendicular lines of a slope is equal to -1. There are no slope forms in vertical lines.
• A positive slope moves in an upward direction and form a positive relationship. Conversely, inverse variables produce a negative slope and move in a downward direction.
• The equation of a line is m = rise/run = tanθ = Δy/Δx = (y₂ - y₁)/(x₂ - x₁).

Read Also: Horizontal and Vertical Lines

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Previous Year Questions

• A plane makes intercepts a, b, c at A, B, C on the coordinate axes… (KEAM)
• (0,−1) and (0, 3) are two opposite vertices of a square. The other… (BITSAT 2006)
• The length of the straight line x - 3y = 1 intercepted by… (VITEEE 2007)
• A straight line makes an intercept on the Y-axis twice as long as that… (TS EAMCET 2017)
• If a line intercepted between the coordinate axes is trisected at a… (JEE Main 2014)
• If the x-intercept of some line L is double that of the line… (JEE Main 2013)
• If m is the slope of one of the lines represented by… (KCET 2010)
• The locus of the point of intersection of lines…

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