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.

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.

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)

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

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).