Finding Slope From Two Points Formula - Cuemath

Finding Slope From Two Points

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.

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.

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.

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