What is the Slope Formula? Formula, Equation & Examples

Lesson Overview

• What Is the Slope Formula?
• Derivation of Slope Formula
• How to Find the Slope of an Equation
• How to Find the Slope of a Line on a Graph?
• Examples Using Slope Formula

Slope essentially describes the steepness and direction of a line. It tells us how much one quantity changes in relation to another.

The applications of slope are numerous and diverse. It's used to design roads, ramps, roofs, and it also helps calculate velocity and acceleration.

What Is the Slope Formula?

The slope formula finds the steepness and direction of a line on a graph. It is calculated by dividing the vertical change (rise) by the horizontal change (run) between two points on the line.

Formula:

Given two points on a line, P₁(x₁, y₁) and P₂(x₂, y₂), the slope m is calculated as:

m = (y₂ - y₁) / (x₂ - x₁) =   ⃤  y/  ⃤   x

Example:

Consider a line passing through points (-1, 2) and (3, 6).

m = (6 - 2) / (3 - (-1)) = 4 / 4 = 1Therefore, the slope of the line is 1, indicating that for every unit increase in the x-coordinate, there is a corresponding unit increase in the y-coordinate.

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Derivation of Slope Formula

We can find the slope using the x and y coordinates of any two points on the line.

• Slope as Rise over Run: Think of slope as how much a line goes up or down (rise, Δy) for every unit it moves across (run, Δx).
• Formula: m = Δy / Δx
  

Slope is also connected to the angle (θ) the line makes with the positive x-axis:

• Slope and Tangent: m = tan θ
  

In a right triangle, tangent is the ratio of the opposite side (height) to the adjacent side (base):

• Tangent in terms of coordinates: tan θ = (y₂ - y₁) / (x₂ - x₁)
  

Combining these ideas gives us the slope formula:

• Slope Formula: m = Δy / Δx = (y₂ - y₁) / (x₂ - x₁)

This formula lets us calculate the slope using the coordinates of two points.

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How to Find the Slope of an Equation

Finding the slope of an equation depends on the form of the equation. Here's a breakdown for the most common form, the slope-intercept form:

1. Slope-Intercept Form

• Equation: y = mx + b
  
  • 'm' represents the slope.
  • 'b' represents the y-intercept (where the line crosses the y-axis).
    
• Finding the Slope: If the equation is in this form, the slope is simply the coefficient of the 'x' term.
• Example:
  • y = 3x - 5

The slope (m) is 3.

2. Converting to Slope-Intercept Form

If the equation is not in slope-intercept form, you'll need to rearrange it:

• Isolate 'y': Manipulate the equation to get 'y' by itself on one side of the equal sign.
  
• Example:

The slope (m) is -2/3.

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How to Find the Slope of a Line on a Graph?

Finding the slope of a line on a graph means measuring how steep it is by comparing how much it rises for every step forward. It's like comparing a ladder's height to the slide's base length.

The angle (𝜽) is the tilt of the slide. A bigger angle means a steeper slide. Tan 𝜽 helps connect the angle of the slide to its slope. It tells the ratio of rise to every step forward.

The slope of a line is defined as m = tan 𝜽 . If two points A(x1, y1) and B(x2, y2) lie on the line where x1 ≠ x2, the slope of the line AB is determined as:

m= tan 𝜽 = y2-y1/ x2-x1

Here, 𝜽 represents the angle that the line AB makes with the positive x-axis. The angle 𝜽 ranges between 0° and 180°.

It is important to note that θ = 90° occurs only when the line is parallel to the y-axis, i.e., at x1 = x2. at this specific angle, the slope of the line is undefined.

For example -
Let's say A (x1, y1) is (1, 3) and B (x2, y2) is (2, 5). We'd plug those numbers into the formula:
m = (5 - 3) / (2 - 1) = 2/1 = 2  

So the slope is 2 here.

Conditions for perpendicularity, parallelism, and collinearity of straight lines are provided below:

Slope for Parallel Lines

Let l1 and l2 be two parallel lines with inclinations α and β, respectively. For lines to be parallel, their inclinations must be identical, i.e., α = β. This means tan α = tan β. Thus, the condition for two lines with inclinations α and β to be parallel is tan α = tan β.

If the slopes of two lines on a Cartesian plane are the same, the lines will be parallel to each other.

Therefore, for parallel lines, m1 = m2.

In general, for n lines, they are parallel if and only if all their slopes are equal.If the equations of the two lines are represented as ax + by + c = 0 and a'x + b'y + c' = 0, the lines will be parallel if

ab' = a'b.

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Slope for Perpendicular Lines

In the figure, two lines l1 and l2 are shown with inclinations α and β. If the lines are perpendicular, then β = α + 90°. (Based on angle properties)

The slopes of the lines can be written as:

m1 = tan(α + 90°) and m2 = tan α.

m1 = -cot α = - (1 / tan α) = - (1 / m2)

m1 = - (1 / m2)

m1 * m2 = -1

Thus, for two lines to be perpendicular, the product of their slopes must equal -1.

If the equations of the two lines are ax + by + c = 0 and a'x + b'y + c' = 0,

they are perpendicular when aa' + bb' = 0. (This result can be derived by calculating the slopes of the lines and setting their product equal to -1.)

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Slope for Collinearity

For the lines AB and BC to be collinear, their slopes must be identical, and they must share at least one common point. Therefore, for points A, B, and C to be collinear, the slopes of the lines AB and BC must be equal.

If the equations of the two lines are given as

ax + by + c = 0 and a'x + b'y + c' = 0, the lines will be collinear when ab'c' = a'b'c = a'c'b.

Angle between Two Lines

When two lines meet at a point, the angle between them can be calculated using their slopes, and it is given by the following formula:

tan θ = |(m2 - m1) / (1 + m1 * m2)|

where m1 and m2 represent the slopes of lines AB and CD, respectively.

If (m2 - m1) / (1 + m1 * m2) is positive, then the angle between the lines is acute.

If (m2 - m1) / (1 + m1 * m2) is negative, then the angle between the lines is obtuse.

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Slope of Vertical Lines

Vertical lines do not have a slope, as they do not exhibit any steepness. In other words, the steepness of vertical lines cannot be defined. A vertical line has no values for its x-coordinates. Therefore, using the formula for the slope of a line:

Slope, m = (y2 – y1)/(x2 – x1)

For vertical lines, x2 = x1 = 0.

Thus, m = (y2 – y1)/0 = undefined.

Similarly, the slope of a horizontal line is 0, as the y-coordinates remain constant.

m = 0/(x2 – x1) = 0 [for horizontal line].

Positive and Negative Slope

If the slope of a line is positive, it indicates that the line rises as we move along the x-axis, meaning the rise over run is positive.

If the slope is negative, the line descends as we move along the x-axis.

Examples Using Slope Formula

Example 1: Find the slope of the line passing through the points (2, 3) and (5, 9).

Solution:

• Label the points:
  • (x₁, y₁) = (2, 3)
  • (x₂, y₂) = (5, 9)
• Substitute the values into the formula:
  • m = (9 - 3) / (5 - 2)
• Simplify:
  • m = 6 / 3 = 2
  • Therefore, the slope of the line is 2.
    

Example 2: Find the slope of the line passing through the points (-1, 4) and (3, -2).
Solution:

• Label the points:
  • (x₁, y₁) = (-1, 4)
  • (x₂, y₂) = (3, -2)
• Substitute the values into the formula:
  • m = (-2 - 4) / (3 - (-1))
• Simplify:
  • m = -6 / 4 = -3/2
  • Therefore, the slope of the line is -3/2.
    

Example 3: A line has a slope of 4 and passes through the point (1, 5). Find another point on the line.

Solution:

• Use the slope formula and the given information:
  • m = 4
  • (x₁, y₁) = (1, 5)
  • Let (x₂, y₂) be the unknown point.
• Choose a value for x₂ (any value will work). Let's say x₂ = 3.
• Substitute the values into the slope formula and solve for y₂:
  • 4 = (y₂ - 5) / (3 - 1)
  • 4 = (y₂ - 5) / 2
  • 8 = y₂ - 5
  • y₂ = 13
• The other point: (x₂, y₂) = (3, 13)
  

Example 4: Determine if the points (0, 1), (2, 3), and (4, 5) are collinear (lie on the same line).

Solution:

• Find the slope between the first two points:
  • m₁ = (3 - 1) / (2 - 0) = 2 / 2 = 1
• Find the slope between the second and third points:
  • m₂ = (5 - 3) / (4 - 2) = 2 / 2 = 1
• Compare the slopes:
  • Since m₁ = m₂, the points are collinear.
    

Example 5: Find the slope of a line that is perpendicular to a line with a slope of -2/3.

Solution:

• Recall that the slopes of perpendicular lines are negative reciprocals of each other.
• Find the negative reciprocal of -2/3:
  • The negative reciprocal is 3/2.
• Therefore, the slope of the perpendicular line is 3/2.

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