### Learning Goals

I will be able to solve problems involving linear relations

Did you know that the United States uses miles to measure distances, while most of the rest of the world uses kilometers? Suppose you were an American visiting Asia, and you were using a graph to convert kilometers to miles to make distances easier to understand. Would you know how to convert a distance such as 10 kilometers? After completing this Concept, you'll be able use graphs to solve problems such as this one.

Most of the world uses kilometres to measure distances. What if you decide to vacation in the United States? Do you know how to convert to miles? How can a graph help you solve these problems?

### Guidance

Analyzing linear graphs is a part of life

– whether you are trying to decide to buy stock,

- figuring out if your blog readership is increasing, or

- predicting the temperature from a weather report

Although linear graphs can be quite complex, such as a six-month stock graph, many are very basic to analyze.

#### Example A

The graph below shows the solutions to the price before tax and the price after tax at a particular store. Determine the price after tax of a $6.00 item.

By finding the appropriate \begin{align*}x\end{align*} value ($6.00), you can find the solution, which is the \begin{align*}y\end{align*} value (approximately $6.80). Therefore, the price after tax of a $6.00 item is approximately $6.80.

#### Example B

The following graph shows the linear relationship between Celsius and Fahrenheit temperatures. Using the graph, convert \begin{align*}70^\circ F\end{align*} to Celsius.

By finding the temperature of \begin{align*}70^\circ F\end{align*} and locating its appropriate Celsius value, you can determine that \begin{align*}70^\circ F \approx 22^\circ C\end{align*}.

#### Example C

Use the same graph to convert 30 degrees Celsius to Farenheit.

By finding the temperature of \begin{align*}30^\circ C\end{align*} and locating its appropriate Fahrenheit value, you can determine that \begin{align*}30^\circ C \approx 85^\circ F\end{align*}.

### Guided Practice

Suppose a job pays $20 per hour. A graph of income based on hours worked is shown below. Use the graph to determine how many hours are required to earn $60.

**Solution:**

By finding the amount of $60 on the vertical axis, you can follow a horizontal line through the value, until it meets the graph. Follow a vertical line straight down from there, until it meets the horizontal axis. There, the value in hours is 3. This means that the number of hours of work needed to earn $60 is 3 hours.

### Practice

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Graphs of Linear Equations (13:09)

- Using the tax graph from the Concept, determine the net cost of an item costing $8.00 including tax.
- Using the temperature graph from the Concept, determine the following:
- The Fahrenheit temperature of \begin{align*}0^\circ C\end{align*}
- The Celsius temperature of \begin{align*}0^\circ F\end{align*}
- The Celsius equivalent to the boiling point of water \begin{align*}(212^\circ F\end{align*})

- At the airport, you can change your money from dollars into Euros. The service costs $5, and for every additional dollar you get 0.7 Euros. Make a table for this information and plot the function on a graph. Use your graph to determine how many Euros you would get if you give the exchange office $50.

The graph below shows a conversion chart for converting between the weight in kilograms and the weight in pounds. Use it to convert the following measurements.

- 4 kilograms into weight in pounds
- 9 kilograms into weight in pounds
- 12 pounds into weight in kilograms
- 17 pounds into weight in kilograms