Finding Slope From Two Points - Formula, Calculating Slope (2024)

Finding slope from two points is just the application of the slope formula rise/run. There are different formulas to find the slope with different types of available information about the line. Finding the slope from two points formula is specifically used when two points on the line are given.

Let us see how to derive the formula for finding the slope from two points and also we will solve a few examples using the formula.

1.Finding Slope From Two Points Formula
2.Calculating Slope From Two Points Derivativation
3.Steps for Finding Slope From Two Points
4.Slope From Two Points Examples
5.FAQs on Finding Slope From Two Points

Finding Slope From Two Points Formula

The formula for finding slope from two points (x₁, y₁) and (x₂, y₂) on a line is m = (y₂ - y₁) / (x₂ - x₁). Here,

  • m = slope of the line
  • x₁ = the x-coordinate of the first point
  • y₁ = the y-coordinate of the first point
  • x₂ = the x-coordinate of the second point
  • y₂ = the x-coordinate of the second point

We know that we find the slope of a line from its graphing by using the formula rise/run. We can use the same formula to derive the above formula also. Consider a line with two points A (x₁, y₁) and B (x₂, y₂) on it.

Then rise from A to B = y₂ - y₁

Run from A to B = x₂ - x₁

Then the slope, m = rise/run = (y₂ - y₁) / (x₂ - x₁)

Hence, we derived the slope formula. We can visualize this in the figure below.

Finding Slope From Two Points - Formula, Calculating Slope (1)

Calculating Slope From Two Points Derivativation

Apart from the method that is already shown above, we can derive the formula of finding slope from two points in different methods. Let us see them. In each of these methods, consider two points A(x₁, y₁) and B(x₂, y₂) on the line.

Method 1

Let θ be the angle made by the line with the positive direction of the x-axis. Draw a horizontal and a vertical line from the two points A and B respectively such that they meet at C.

Finding Slope From Two Points - Formula, Calculating Slope (2)

By the corresponding angles property, the angle at A = θ. By applying tan to the triangle ABC,

tan θ = (Opposite)/(Adjacent)

tan θ = (y₂ - y₁) / (x₂ - x₁) ... (1)

We know that if θ is the angle made by a straight line with the positive direction of x-axis then its slope is,

m = tan θ ... (2)

From (1) and (2),

m = (y₂ - y₁) / (x₂ - x₁)

Method 2

We know that the slope-intercept form of a line is y = mx + b.

  • Since A(x₁, y₁) lies on the line, y₁ = mx₁ + b ... (3)
  • Since B(x₂, y₂) lies on the line, y₂ = mx₂ + b ... (4)

We will solve (3) and (4) by the substitution method. Frm (3), b = y₁ - mx₁. Substituting this in (4):

y₂ = mx₂ + y₁ - mx₁

Subtracting y₁ from both sides,

y₂ - y₁ = mx₂ - mx₁

Taking m as common factor on the right side,

y₂ - y₁ = m(x₂ - x₁)

Dividing both sides by x₂ - x₁,

m = (y₂ - y₁) / (x₂ - x₁)

Steps for Finding Slope From Two Points

Here are the steps to find the slope of a line given two points on it.

  • In the first point, denote the x coordinate with x₁ and denote the y-coordinate with y₁.
  • In the second point, denote the x coordinate with x₂ and denote the y-coordinate with y₂.
  • Find the differences y₂ - y₁ and x₂ - x₁.
  • Divide the difference of y-coordinates by the difference of x-coordinates to find the slope (m). i.e., m = (y₂ - y₁) / (x₂ - x₁).

Note: We can interchange the points while finding slope without affecting the answer.

Example: Find the slope of the line passing through the points (1, -2) and (3, -6).

Solution:

Let (1, -2) = (x₁, y₁)

and (3, -6) = (x₂, y₂)

Then x₁ = 1, y₁ = -2, x₂ = 3, and y₂ = -6.

Slope, m = (y₂ - y₁) / (x₂ - x₁)
= (-6 - (-2)) / (3 - 1)
= (-6 + 2) / (3 - 1)
= (-4) / 2
= -2

So the slope of the given line is -2.

Important Notes on Calculating Slope From Two Points:

  • The slope of a line given two points (x₁, y₁) and (x₂, y₂) can be calculated
    either using m = (y₂ - y₁) / (x₂ - x₁)
    or using m = (y₁ - y₂) / (x₁ - x₂)
  • The order that we follow should be the same in the numerator and the denominator. i.e., it cannot be something like (y₁ - y₂) / (x₂ - x₁).
  • We get the same slope for a line irrespective of which two points on it we are using.
  • The slope of a line is 0 if the difference in y-coordinates is 0 and in this case, the line is horizontal.
  • The slope of a line is not defined if the difference in x-coordinates is 0 and in this case, the line is vertical.

☛ Related Topics:

  • Finding Slope From Two Points Calculator
  • Point Slope Form Calculator
  • Slope Intercept Form Calculator
  • Equation of Line Calculator

FAQs on Finding Slope From Two Points

What is the Formula for Finding Slope From Two Points?

For finding slope from two points of a line (x₁, y₁) and (x₂, y₂), we use the formula (y₂ - y₁) / (x₂ - x₁). i.e., it is the ratio of difference of y-coordinates to the difference of x-coordinates such that the differences are calculated in the same order.

How to Find the Equation of Line with Two Points?

To find the equation of a line with two points (x₁, y₁) and (x₂, y₂):

  • Compute its slope using the formula, m = (y₂ - y₁) / (x₂ - x₁)
  • Use the point-slope form of a line formula to find the equation which is: y - y₁ = m(x₂ - x₁).

Can we Use (y1-y2)/(x1-x2) to Calculate Slope of a Line From Two Points?

If (x₁, y₁) and (x₂, y₂) are two points on a line, then the usual formula we use to compute the slope is (y₂ - y₁) / (x₂ - x₁). But if we take -1 as a common factor from both numerator and denominator, then -1 gets canceled and we will be left with (y₁ - y₂) / (x₁ - x₂). So this formula is also applicable for finding the slope of a line with two points.

How to Find the Slope of a Line Passing Through Two Points?

If we consider two points (x₁, y₁) and (x₂, y₂) on a line where the first point lies lower then than the upper point on an increasing line, then the rise = y₂ - y₁ and run = x₂ - x₁. Then the slope = rise/run = (y₂ - y₁) / (x₂ - x₁). This is the formula we use to find the slope of a line passing through two given points.

How to Write Slope-Intercept Form Using Two Points?

To find the equation of line in slope-intercept form when two points (x₁, y₁) and (x₂, y₂) on a line are given,

  • Find the slope using m = (y₂ - y₁) / (x₂ - x₁).
  • Substitute it in the equation y = mx + b.
  • Substitute any one of the given points to be (x, y) in the above equation and solve for b.
  • Substitute the value of b back into the same equation (from the second step).

How to Find Slope of a Line From a Graph?

To find the slope of a line when it is shown on a graph:

  • Identify one random point on it and denote it by (x₁, y₁).
  • Identify another point on it and denote it by (x₂, y₂).
  • Apply the formula, (y₂ - y₁) / (x₂ - x₁).
Finding Slope From Two Points - Formula, Calculating Slope (2024)

FAQs

Finding Slope From Two Points - Formula, Calculating Slope? ›

To find the slope between two points, we use the slope-point formula. If we have two points, (x 1,y 1) and (x 2,y 2), the point slope formula is given by: m=(y 2—y 1)/(x 2—x 1). m is the variable used to represent slope. We plug in our specific x and y values and simplify to find the slope.

How do you find the slope of the line that passes through the points 2? ›

To find the slope between two points, we use the slope-point formula. If we have two points, (x 1,y 1) and (x 2,y 2), the point slope formula is given by: m=(y 2—y 1)/(x 2—x 1). m is the variable used to represent slope. We plug in our specific x and y values and simplify to find the slope.

How do you find the answer to a slope? ›

Find slope by finding the difference in the y points, and divide that by the difference in the x points.
  1. The difference between y coordinates Δy is.
  2. The difference between x coordinates Δx is.
  3. Divide Δy by Δx to find slope m.
Feb 6, 2024

How to find a line with 2 points? ›

Given two points on a line, we can write an equation for that line by finding the slope between those points, then solving for the y-intercept in the slope-intercept equation y=mx+b. In this example, we write an equation of the line that passes through the points (-1,6) and (5,-4).

How to calculate for slope? ›

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)

What are the 2 equations for slope? ›

FactYou can use this fact when you know:
Slope formula:two points on a line
Slope-intercept formula: y = mx + bthe slope and y-intercept of a line
Point-slope formula: y - y1 = m(x - x1)the slope of a line and a point on the line
Parallel lines have equal slopesthe slope of a line
1 more row

What is the point-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). Watch this video to learn more about it and see some examples. Created by Sal Khan.

What is the equation of a line passing through two points with slope? ›

Finding the Equation of the Line Passing through Two Given Points
  • Find the slope/gradient of the line.
  • Substitute the values of the slope and any one of the given points into the formula.
  • Simplify to obtain an equation resembling the standard equation of the line, i.e., Ax + By + C = 0, where A, B, and C are constants.

What is an example of a slope? ›

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.

How to find slope from 2 points? ›

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.

How to find slope-intercept form from two points? ›

How to Find the Equation of a Line from Two Points
  1. Find the slope using the slope formula. ...
  2. Use the slope and one of the points to solve for the y-intercept (b). ...
  3. Once you know the value for m and the value for b, you can plug these into the slope-intercept form of a line (y = mx + b) to get the equation for the line.

What is the formula for the slope function? ›

In the case of a straight line y=mx+b, the slope m=Δy/Δx measures the change in y per unit change in x. This can be interpreted as a measure of "sensitivity''; for example, if y=100x+5, a small change in x corresponds to a change one hundred times as large in y, so y is quite sensitive to changes in x.

What is the point 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). Watch this video to learn more about it and see some examples. Created by Sal Khan.

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