Suppose a person's annual salary is directly proportional to the number of years he or she has spent in school. What do you think this means? Could you determine the constant of proportionality? What information would you need to do so? If you wrote a linear equation to represent this relationship, what would be the slope and \begin{align*}y-\end{align*}intercept?

### Direct Variation

At the local farmer’s market, you saw someone purchase 5 pounds of strawberries and pay $12.50. You want to buy strawberries too, but you want only 2 pounds. How much would you expect to pay?

This situation is an example of a **direct variation**. You would expect that the strawberries are priced on a “per pound” basis, and that if you buy two-fifths of the amount of strawberries, you would pay two-fifths of $12.50 for your strawberries, or $5.00. Similarly, if you bought 10 pounds of strawberries (twice the amount), you would pay $25.00 (twice $12.50), and if you did not buy any strawberries you would pay nothing.

**Direct variation** can be expressed as the equation \begin{align*}y=(k)x\end{align*}, where \begin{align*}k\end{align*} is called the constant of proportionality.

Direct variation occurs when:

- The fraction \begin{align*}\frac{rise}{run}\end{align*} or \begin{align*}\frac{change \ in \ y}{change \ in \ x}\end{align*} is always the same, and
- The ordered pair (0, 0) is a solution to the equation.

#### Let's use direct variation to solve the following problem:

If \begin{align*}y\end{align*} varies directly with \begin{align*}x\end{align*} according to the relationship \begin{align*}y=k \cdot x\end{align*}, and \begin{align*}y=7.5\end{align*} when \begin{align*}x=2.5\end{align*}, determine the constant of proportionality, \begin{align*}k\end{align*}.

We can solve for the constant of proportionality using substitution.

Substitute \begin{align*}x=2.5\end{align*} and \begin{align*}y=7.5\end{align*} into the equation \begin{align*}y=k \cdot x\end{align*}.

\begin{align*}7.5 & = k(2.5) && \text{Divide both sides by} \ 2.5.\\ \frac{7.5}{2.5}& =k=3\end{align*}

The constant of variation (or the constant of proportionality) is 3.

You can use this information to graph this direct variation situation. Remember that all direct variation situations cross the origin. You can plot the ordered pair (0, 0) and use the constant of variation as your slope.

#### Now, explain why each of the following equations are not examples of direct variation:

\begin{align*}y& =\frac{2}{x}\\ y& =5x-1\\ 2x+y& =6\end{align*}

In equation 1, the variable is in the denominator of the fraction, violating the definition.

In equation 2, there is a \begin{align*}y-\end{align*}intercept of –1, violating the definition.

In equation 3, there is also a \begin{align*}y-\end{align*}intercept, violating the definition.

#### Translating a Sentence into a Direct Variation Equation

Direct variation equations use the same phrase to give the reader a clue. The phrase is either “directly proportional” or “varies directly.”

#### Let's translate the following sentence into an equation:

The area of a square varies directly as the square of its side.

The first variable you encounter is “area.” Think of this as your \begin{align*}y\end{align*}. The phrase “varies directly” means \begin{align*}= (k)\times\end{align*}. The second variable is “square of its side.” Call this letter \begin{align*}s\end{align*}.

Now translate into an equation: \begin{align*}y=(k)\times s^2\end{align*}.

You’ve written your first direct variation equation.

### Example

#### Example 1

Earlier, you were asked to suppose that a person's annual salary is directly proportional to the number of years he or she has spent in school. What does this mean? Could you determine the constant of proportionality? What would the slope and \begin{align*}y-\end{align*}intercept be?

That a person's salary is directly proportional to the number of years that they have spent in school means that as the number of years a person is in school increases, so does the person's annual salary. The increase is constant and described by the constant of proportionality. This would be found by calculating the slope between two data points relating to salary and years in school. The slope in an equation describing this situation would be the constant of proportionality and the \begin{align*}y-\end{align*}intercept would be the salary of a person with 0 years of school.

#### Example 2

The distance you travel is directly proportional to the time you have been traveling. Write this situation as a direct variation equation.

The first variable is distance; call it \begin{align*}d\end{align*}. The second variable is the time you have been traveling; call it \begin{align*}t\end{align*}. Apply the direct variation definition:

\begin{align*}d=(k) \times t\end{align*}

### Review

- Describe direct variation.
- What is the formula for direct variation? What does \begin{align*}k\end{align*} represent?

Translate the following direct variation situations into equations. Choose appropriate letters to represent the varying quantities.

- The amount of money you earn is directly proportional to the number of hours you work.
- The weight of an object on the Moon varies directly with its weight on Earth.
- The volume of a gas is directly proportional to its temperature in Kelvin.
- The number of people served varies directly with the amount of ground meat used to make burgers.
- The amount of a purchase varies directly with the number of pounds of peaches.

Explain why each equation is not an example of direct variation.

- \begin{align*}\frac{4}{x}=y\end{align*}
- \begin{align*}y=9\end{align*}
- \begin{align*}x=-3.5\end{align*}
- \begin{align*}y=\frac{1}{8} x+7\end{align*}
- \begin{align*}4x+3y=1\end{align*}

Graph the following direct variation equations.

- \begin{align*}y=\frac{4}{3}x\end{align*}
- \begin{align*}y=-\frac{2}{3}x\end{align*}
- \begin{align*}y=-\frac{1}{6}x\end{align*}
- \begin{align*}y=1.75x\end{align*}
- Is \begin{align*}y=6x-2\end{align*} an example of direct variation? Explain your answer.

**Mixed Review**

- Graph \begin{align*}3x+4y=48\end{align*} using its intercepts.
- Graph \begin{align*}y=\frac{2}{3} x-4\end{align*}.
- Solve for \begin{align*}u: 4(u+3)=3(3u-7)\end{align*}.
- Are these lines parallel? \begin{align*}y=\frac{1}{2} x-7\end{align*} and \begin{align*}2y=x+2\end{align*}
- In which quadrant is (–99, 100)?
- Find the slope between (2, 0) and (3, 7).
- Evaluate if \begin{align*}a=-3\end{align*}and\begin{align*}b=4\end{align*}: \begin{align*}\frac{1+4b}{2a-5b}\end{align*}.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 4.11.