**(x**and

_{1}, y_{1})**(x**being joined by a line:The equation of the line between the points is given by the formula:

_{2}, y_{2})**y**and**x**are the**Cartesian coordinates**of points on the line.**(x**and_{1}, y_{1})**(x**are two_{2}, y_{2})**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 x_{1}= 1 and y_{1}= 2. - The second point is
**(4, 8)**, so x_{2}= 4 and y_{2}= 8.

## 3

Substitute x

_{1}, y_{1}, x_{2}and y_{2}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)**.

## Why Does the Formula Work?

The equation of a line between two points in made up of two equations. Start with the slope-point form of a linear equation: Then replace the slope**m**with the formula for the slope between two points:

## Alternative Formulas

It does not matter which of the two points is used to be subtracted from the**x**and the

**y**in the formula. The following forms of the equation can both be used:

## Beware

## When Points Have Negative Coordinates

In the formula, the y-coordinate and the x-coordinate are subtracted. What if a Cartesian coordinate is negative, for example (−1, −2)?
y

Remember, that subtracting a negative number is the same as adding the positive number:
**− −2**= ...(x**− −1**)
y

**+ 2**= ...(x**+ 1**)## Interactive Widget

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