Equation of a Line Between Two Points

What Is the Equation of a Line Between Points?

A linear equation is an equation that represents a line.

If we are given two points, we can draw a line between them.

If we know the Cartesian coordinates of the two points, then we can find the equation of the line.

The image below shows what we mean by two points (x1, y1) and (x2, y2) being joined by a line:

The equation of the line between the points is given by the formula:

  • y and x are the Cartesian coordinates of points on the line.

  • (x1, y1) and (x2, y2) are two points on the line.

How to Find the Equation of a Line Between Two Points

Finding the equation of a line between two points is easy.

Question

What is the equation of the line that passes through the points (1, 2) and (4, 8)?

Step-by-Step:

1

Start with the formula:

$$y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$$

2

Find the Cartesian coordinates of the points. In our example:

  • The first point is (1, 2), so x1 = 1 and y1 = 2.

  • The second point is (4, 8), so x2 = 4 and y2 = 8.

3

Substitute x1, y1, x2 and y2 into the formula.

$$y - 2 = \frac{8 - 2}{4 - 1}(x - 1)$$

4

Tidy up the equation.

$$y - 2 = \frac{6}{3}(x - 1)$$ $$y - 2 = 2(x - 1)$$

Answer:

The equation of the line between the points (1, 2) and (4, 8) is y − 2 = 2(x − 1).

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The slider below gives another example of finding the equation of a line passing through two points

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See Also

What is a linear equation? What are Cartesian coordinates? What is the x-coordinate? What is the y-coordinate? What is a linear equation in slope-point form? What is a negative number?