You decide to buy a laptop for $800. In 3 years, the laptop will be worth $450. How much will the computer be worth after 6 years?

Writing a linear equation that relates the two prices will help you determine how much the computer will be worth after 6 years.

### Finding the Equation of a Line

You know how to find the slope between two points. We will now find the entire equation of a line. Recall from Algebra I that the equation of a line in slope-intercept form is \begin{align*}y = mx + b,\end{align*} where \begin{align*}m\end{align*} is the slope and \begin{align*}b\end{align*} is the \begin{align*}y-\end{align*}intercept. You can find the slope either by using slope triangles or the Slope Formula. To find the \begin{align*}y-\end{align*}intercept, or \begin{align*}b,\end{align*} you can either locate where the line crosses the \begin{align*}y-\end{align*}axis (if given the graph) or by using algebra.

Let's find the equation for the following lines.

- Find the equation of the line below.

Analyze the line. We are given two points on the line, one of which is the \begin{align*}y-\end{align*}intercept. From the graph, it looks like the line passes through the \begin{align*}y-\end{align*}axis at (0, 4), making \begin{align*}b = 4\end{align*}. Now, we need to find the slope. You can use slope triangles or the Slope Formula. Using slope triangles, we have:

From this, we see that the slope is \begin{align*}- \frac{2}{6}\end{align*} or \begin{align*}- \frac{1}{3}\end{align*}.

Plugging our found information into the slope-intercept equation, the equation of this line is \begin{align*}y = - \frac{1}{3} x + 4\end{align*}.

Alternate Method: If we had used the Slope Formula, we would use (0, 4) and (6, 2), which are the values of the given points.

\begin{align*}m = \frac{2-4}{6-0} = \frac{-2}{6} = - \frac{1}{3}\end{align*}

The slope of a line is -4 and the \begin{align*}y-\end{align*}intercept is (0, 3). What is the equation of the line?

This problem explicitly tells us the slope and \begin{align*}y-\end{align*}intercept. The slope is -4, meaning \begin{align*}m = -4\end{align*}. The \begin{align*}y-\end{align*}intercept is (0, 3), meaning \begin{align*}b = 3\end{align*}. Therefore, the equation of the line is \begin{align*}y = -4x + 3\end{align*}.

- The slope of a line is \begin{align*}\frac{1}{2}\end{align*} and it passes through the point (4, -7). What is the equation of the line?

In this problem, we are given \begin{align*}m\end{align*} and a point on the line. The point, (4, -7) can be substituted in for \begin{align*}x\end{align*} and \begin{align*}y\end{align*} in the equation. We need to solve for the \begin{align*}y-\end{align*}intercept, or \begin{align*}b\end{align*}. Plug in what you know to the slope-intercept equation.

\begin{align*}y &= mx + b\\ -7 &=\frac{1}{2}(4) + b\\ -7 &=2 + b\\ -9 &=b\end{align*}

From this, the equation of the line is \begin{align*}y = \frac{1}{2}x - 9\end{align*}.

We can test if a point is on a line or not by plugging it into the equation. If the equation holds true, the point is on the line. If not, then the point is not on the line.

- Find the equation of the line that passes through (12, 7) and (10, -1).

In this problem, we are not given the slope or the \begin{align*}y-\end{align*}intercept. First, we need to find the slope using the Slope Formula.

\begin{align*}m = \frac{-1-7}{10-12} = \frac{-8}{-2} = 4\end{align*}

Now, plug in one of the points for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}. It does not matter which point you choose because they are both on the line.

\begin{align*}7 &= 4(12) + b\\ 7 &= 48 + b \\ -41 &= b\end{align*}

The equation of the line is \begin{align*}y = 4x - 41\end{align*}.

### Examples

#### Example 1

Earlier, you were asked to find how much the computer will be worth after 6 years.

To determine the equation of the line, rewrite the given information as points. The first could be (0, 800) and the second would be (3, 450). We already know that the *y*-intercept is 800 because the *x*-value is zero at that point. Find the slope.

\begin{align*}\frac{800-450}{0-3}= - \frac{350}{3}\end{align*}

Therefore, the equation of the declining value of the laptop is \begin{align*}y= - \frac{350}{3}x + 800\end{align*}. In 6 years, the laptop will be worth \begin{align*}y= - \frac{350}{3}\cdot 6 + 800 = -700+800 = 100\end{align*}. The laptop will be worth $100.

#### Example 2

What is the equation of the line where the slope is 1 and passes through (5, 3)?

We are told that \begin{align*}m = 1, x = 5,\end{align*} and \begin{align*}y = 3\end{align*}. Plug this into the slope-intercept equation and solve for \begin{align*}b\end{align*}.

\begin{align*}3 &= 1(5) + b\\ 3 &= 5 + b \\ -2 &= b\end{align*}

The equation of the line is \begin{align*}y = x - 2\end{align*}

#### Example 3

Find the equation of the line that passes through (9, -4) and (-1, -8).

First, find the slope.

\begin{align*}m = \frac{-8-(-4)}{-1-9} = \frac{-4}{-10} = \frac{2}{5}\end{align*}

Now, find the \begin{align*}y-\end{align*}intercept. We will use the second point. Remember, it does not matter which point you use.

\begin{align*}-8 &= \frac{2}{5}(-1) + b\\ -8 &= - \frac{2}{5} + b \\ -7 \frac{3}{5} &= b\end{align*}

The equation of the line is \begin{align*}y = \frac{2}{5}x - 7 \frac{3}{5}\end{align*} or \begin{align*}y = \frac{2}{5}x - \frac{38}{5}\end{align*}.

When your \begin{align*}y-\end{align*}intercept is a fraction, make sure it is reduced. Double-check with your teacher on how s/he wants you to leave your answer.

#### Example 4

Find the equation of the line below.

We can find the slope one of two ways: using slope triangles or by using the Slope Formula. We are given (by the drawn points in the picture) that (-2, 2) and (4, -2) are on the line. Drawing a slope triangle, we have:

We have that the slope is \begin{align*}- \frac{4}{6}\end{align*} or \begin{align*}- \frac{2}{3}\end{align*}. To find the \begin{align*}y-\end{align*}intercept, it looks like it is somewhere between 0 and 1. Take one of the points and plug in what you know to the slope-intercept equation.

\begin{align*}2 &= - \frac{2}{3}(-2) + b\\ 2 &= \frac{4}{3} + b \\ \frac{2}{3} &= b\end{align*}

The equation of the line is \begin{align*}y = - \frac{2}{3}x + \frac{2}{3}\end{align*}.

### Review

Find the equation of each line with the given information below.

- slope = 2, \begin{align*}y-\end{align*}intercept = (0, 3)
- \begin{align*}m = -\frac{1}{4}, \ b = 2.6\end{align*}
- slope = -1, \begin{align*}y-\end{align*}intercept = (0, 2)
- \begin{align*}x-\end{align*}intercept = (-2, 0), \begin{align*}y-\end{align*}intercept = (0, -5)
- slope \begin{align*}= \frac{2}{3}\end{align*} and passes through (6, -4)
- slope \begin{align*}= - \frac{3}{4}\end{align*} and passes through (-2, 5)
- slope = -3 and passes through (-1, -7)
- slope = 1 and passes through (2, 4)
- passes through (-5, 4) and (1, 1)
- passes through (5, -1) and (-10, -10)
- passes through (-3, 8) and (6, 5)
- passes through (-4, -21) and (2, 9)

For problems 13-16, find the equation of the lines in the graph below.

- Green Line
- Blue Line
- Red Line
- Purple Line
- Find the equation of the line with zero slope and passes through (8, -3).
- Find the equation of the line with zero slope and passes through the point (-4, 5).
- Find the equation of the line with zero slope and passes through the point \begin{align*}(a, b)\end{align*}.
**Challenge**Find the equation of the line with an*undefined*slope that passes through \begin{align*}(a, b)\end{align*}.

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 2.2.