**Introduction**

In this lesson, you will learn how to find the equation of a line using two given points.

This video illustrates the lesson material below. Watching the video is optional.

**How to Find the Equation of a Line From Two Points**

Here are two points: \((-5, 13)\) and \((3, -3)\). Both of these points are on the same line (see Figure 1). To find the equation of this line, follow three steps:

Figure 1

**Step 1**

Find the slope using the slope formula: \(m=\frac{y_{2-}y_{1}}{x_{2-}x_{1}}\)

\begin{align*} m&=\frac{y_{2} - y_{1}}{x_{2} - x_{1}} &\color{red}\small\text{Slope formula}\\\\

m&=\frac{13 - (-3)}{-5 - 3} &\color{red}\small\text{Substitute values of x & y}\\\\ m&=\frac{16}{-8}&\color{red}\small\text{Simplify numerator and denominator}\\\\ m&=-2 &\color{red}\small\text{Simplify the fraction}\\\ \end{align*}

The slope is -2 for this equation.

Figure 2

**Step 2**

Use the slope and one of the points to solve for the y-intercept using the slope-intercept form: \(y=mx + b\). It does not matter which point you use to find the equation. Here, \((3, -3)\) is used with \(m = -2\).

\begin{align*} y&=mx+b&\color{red}\small\text{Slope-intercept form}\\\\

-3&=-2(3)+b&\color{red}\small\text{Substitute values of m, x, & y}\\\\ -3&= -6 +b &\color{red}\small\text{Right side: multiply -2 & 3}\\\\ -3 \color{red}\text{+6} &= -6 + b \color{red}\text{+6} &\color{red}\small\text{Add 6 to both sides} \\\\ -3 \color{red}\text{+6} &= b &\color{red}\small\text{Right side: the 6's will cancel} \\\\ 3&= b &\color{red}\small\text{Left side: subtract 3 from 6} \end{align*}

The *y*-intercept, or *b*, is equal to 3.

**Step 3**

Once you know the value for *m* and the value for *b*, you can substitute these into the slope-intercept form to get the equation for the line. The equation of the line that both of these points go through is \(y=-2x+3\).

**Things to Remember**

- The slope formula is \(m=\frac{y_{2-}y_{1}}{x_{2-}x_{1}}\).
- The steps to find the equation of a line, when given two points are:
- Find the slope using the slope formula.
- Use the slope and one of the points given to solve for the y-intercept in the slope-intercept form of a line: \(y=mx + b\).
- Use the slope (
*m*) and y-intercept (*b*) to write the equation of the line using the slope-intercept form.

### Practice Problems

**For each of the following problems, find the equation of the line that passes through the following two points:**

- \(\left ( -5,10 \right )\) and \(\left ( -3,4 \right )\) (Solution
- \(\left ( -5,-26 \right )\) and \(\left ( -2,-8 \right )\) (Solution
- \(\left ( -4,-22 \right )\) and \(\left ( -6,-34 \right )\) (Solution
- \(\left ( 3,1 \right )\) and \(\left ( -6,-2 \right )\) (Solution
- \(\left ( 4,-6 \right )\) and \(\left ( 6,3 \right )\) (Solution
- \(\left ( 5,5 \right )\) and \(\left ( 3,2 \right )\) (Solution

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