Slope Formula - What is Slope Formula? Equation, Examples (2024)

The slope formula is used to calculate the inclination or steepness of a line. It finds application in determining the slope of any line by finding the ratio of the change in the y-axis to the change in the x-axis. The slope of a line is defined as the change in the "y" coordinate with respect to the change in the "x" coordinate of that line.

What is the Slope Formula?

The slope formula refers to the formula used to calculate the steepness of a line and determines how much it's inclined. To calculate the slope of the lines, the x and y coordinates of the points lying on the line can be used. In other words, it is the ratio of the change in the y-axis to the change in the x-axis.

Slope Formula - What is SlopeFormula? Equation, Examples (1)

Slope Formula

The formula to calculate slope is given as,

m = (y2 - y1)/(x2 - x1) = Δy/Δx

where, m is the slope of the line,x1, x2 are the coordinates of the x-axis, and y1, y2 are the coordinates of the y-axis.

Derivation of Slope Formula

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,

m = Δy/Δx

where

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

Slope Formula - What is SlopeFormula? Equation, Examples (2)

We know that tan θ is also the slope of the line where θ is the angle made by the line with the positive direction of the x-axis.

And, tanθ = height/base

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

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

From the graph, we observe:

  • Δy = (y2 - y1)
  • Δx = (x2 - x1)

Thus, the slope formula is given as: Slope = m = (y2 - y1)/(x2 - x1)

Slope Equation

As we discussed in the previous section, the slope formula can be used to determine the slope of any line. The equation that can be used in finding this slope can therefore written as,
m = rise/run = tanθ = Δy/Δx = (y2 - y1)/(x2 - x1)

where,

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

Also, the equation of slope of any line using the line equation can be given as,
y = mx + b

where,

  • m is the slope of the line
  • b is the y-intercept of the line

Let us see the applications of the slope formula in the following solved examples.

Slope Formula - What is SlopeFormula? Equation, Examples (3)

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Examples Using Slope Formula

Example 1: Find the slope of a line whose coordinates are (2,9) and (4,1)?

Solution:

To find: The slope of the line with coordinates (2,9) and (4,1)

Given, (x1, y1) = (2, 9) and (x2, y2) = (4, 1)

The slope formula is m = (y2 - y1)/(x2 - x1)

m = (1 − 9)/(4 − 2)

m = -8/2 = -4

Answer: Slope of the given line = -4

Example 2: Determine the value of b, if the slope of a line passing through the points (b, 7) and (8, -5) is 6.

Solution:

To find: the value of b

Given, Slope = m = 6, Points: (x1, y1) = (b, 7) and (x2, y2) = (8, -5)
We know that Slope (m) = (y2 - y1)/(x2 - x1)
6 = (-5-7)/(8-b)
6 = (-12)/(8-b)
-2= (8-b)
-2-8 = -b,
b = 10
Answer: The value of b = 10.

Example 3: If the angle made by a line with the positive y−axis is 30°, then what is the value of the slope of the line?

Solution:

To find: slope of the line

Given: Angle made by a line with the positive y−axis = 30°

We know that if the line makes an angle of 30° from the positive y-axis, then it makes an angle of 120° from the positive x-axis. Therefore, the value of the slope of the line is tan120° = -√3

Answer: The value of the slope of the line = -√3

FAQs on Slope Formula

What is the Slope Formula in Geometry?

In geometry, the slope formula is defined as the formula to calculate the slope of any line by finding the ratio of the change in the y-axis to the change in the x-axis.

What does m Stand For in Slope Formula?

In the slope formula, m stands for slope which is calculated as the m = (y2 - y1)/(x2 - x1) = Δy/Δx

What is the Slope Formula Used For?

The slope formula is used to calculate the steepness of a line and the x and y coordinates of the lines are used for the same. We know that the slope of a line is one of the most important characteristics of a line as it helps to measure the rate of change.

How to Calculate Slope Using Slope Formula?

The slope of a line can be calculated using the following formulas:

When coordinates are given:

  • Step 1: Find the coordinates of the line.
  • Step 2: Put their values in the formula, (m) = (y2 - y1)/(x2 - x1)

When angle is given:

  • Step 1: Identify the angle made with the axis.
  • Step 2: Put the value in the formula, m = tanθ

What is the Equation of Slope of a Line?

The equation of slope of any line can be calculated using the following formula,

m = rise/run = tanθ = Δy/Δx = (y2 - y1)/(x2 - x1)

where,

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

The slope equation using the equation of line is given as, y = mx + b, here, m is the slope and b is the y-intercept.

Slope Formula - What is Slope Formula? Equation, Examples (2024)

FAQs

Slope Formula - What is Slope Formula? Equation, Examples? ›

The slope, or steepness, of a line is found by dividing the vertical change (rise) by the horizontal change (run). The formula is slope =(y₂ - y₁)/(x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are the coordinates of two points on the line. Created by Sal Khan and Monterey Institute for Technology and Education.

What is a slope equation example? ›

The equation of the line is written in the slope-intercept form, which is: y = mx + b, where m represents the slope and b represents the y-intercept. In our equation, y = 6x + 2, we see that the slope of the line is 6.

What is the slope formula for given equation? ›

In mathematical terms, the slope is the rate of change of y with respect to x. When dealing with linear equations, we can easily identify the slope of the line represented by the equation by putting the equation in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.

How to do slope step by step? ›

In order to calculate the slope of a line:
  1. Select two points on the line.
  2. Sketch a right angled triangle and label the change in y and the change in x.
  3. Divide the change in y by the change in x to find m.

What is the simple slope equation? ›

The slope equation using the equation of line is given as, y = mx + b, here, m is the slope and b is the y-intercept.

What is the slope formula in words? ›

The slope of a line is a measure of its steepness. Mathematically, slope is calculated as "rise over run" (change in y divided by change in x).

How do you teach the slope formula? ›

Try thinking about the formula of how to find the slope of a line. Formula: Slope = The change in y divided by the change in x.

What are the three slope formulas? ›

Forms of linear equations review
Slope-interceptPoint-slopeStandard
y = m x + b ‍y − y 1 = m ( x − x 1 ) ‍A x + B y = C ‍
where ‍ is slope and ‍ is the ‍ -interceptwhere ‍ is slope and ( x 1 , y 1 ) ‍ is a point on the linewhere ‍ , ‍ , and ‍ are constants

What is the point to slope formula? ›

Point-slope is the general form y-y₁=m(x-x₁) for linear equations. It emphasizes the slope of the line and a point on the line (that is not the y-intercept). We can rewrite an equation in point-slope form to be in slope-intercept form y=mx+b, to highlight the same line's slope and y-intercept.

How is the slope formula written? ›

The graph of the linear equation y = mx + c is a line with m as slope, m and c as the y-intercept. This form of the linear equation is called the slope-intercept form, and the values of m and c are real numbers. The slope, m, represents the steepness of a line.

What is the equation for slope form? ›

Slope-intercept form (y=mx+b) of linear equations highlights the slope (m) and the y-intercept (b) of a line. Watch this video to learn more about it and see some examples.

How to find the equation of a line? ›

To find an equation of a line given the slope and a point.
  1. Identify the slope.
  2. Identify the point.
  3. Substitute the values into the point-slope form, y − y 1 = m ( x − x 1 ) . y − y 1 = m ( x − x 1 ) .
  4. Write the equation in slope-intercept form.
May 6, 2020

What is an example of a slope calculation? ›

For example, the equation y - 3 = 2(x - 1) is in the point-slope form and the slope is 2 since the 2 is in the m spot. The equation y = 3x + 4 is in the slope-intercept form with a slope of 3 since the 3 is in the m spot. For parallel lines, the slope is the same.

What does the slope formula mean? ›

The slope formula definition is a mathematical formula used to calculate the steepness of a line. s l o p e = y 2 − y 1 x 2 − x 1. In this formula, x and y represent coordinates or points on a graph.

How do you write an equation in slope? ›

Learn how to write an equation in slope-intercept form (y=mx+b) for the line with a slope of -3/4 that goes through the point (0,8). We identify the slope (m) and y-intercept (b) to create our equation y = (-3/4)*x + 8.

What is slope easy examples? ›

When plotting a line on a graph, the “Rise” refers to the change in y that corresponds to a specific change in x. This change in x is called the “Run.” For instance, if y increases by 4 when x increases by 2, then Rise = 4 and Run = 2. To find the slope, divide 4/2 to get 2.

What is a real example of slope? ›

The concept of slope has many applications in the real world. In construction, the pitch of a roof, the slant of the plumbing pipes, and the steepness of the stairs are all applications of slope. We can assign a numerical value to the inclination of a line by finding the ratio of the rise and run. This is the slope.

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