Application of Lines

Applying Direct Variation

Direct variation can be used to model real-world situations.

Many real-world situations can be modeled by direct variation. In geometry, the perimeter PP of a square is proportional to the side length ss of the square by the formula P=4sP=4s. The formula d=rtd=rt describes how distance dd is proportional to time tt by the constant of proportionality rr, which represents a constant rate of travel. When shopping for several same-price items, the total cost CC is proportional to the number of items nn by the price pp of each item, or C=pnC = pn.

Sales tax is another example of direct variation. The amount of sales tax on an item is proportional to the price. If the sales tax rate is rr, then the amount of sales tax on an item with price xx is rxrx. The total cost yy of the item with tax is:
y=x+rxy=(1+r)x\begin{aligned}y&=x + rx\\y&=(1+r)x\end{aligned}
Another application of direct variation is converting units, such as distance, volume, or currency. The constant of variation is the conversion factor.
Step-By-Step Example
Applying Direct Variation to Sales Tax
Graph and then determine the exact total cost of an item with a price of $5 after a sales tax of 10%.
Step 1
The total cost yy of an item with price xx after a sales tax rate of rr is represented by the sales tax equation:
y=(1+r)xy=(1 + r)x
For a 10% sales tax, the value of rr is 0.10, which can also be written as 0.1. Substitute 0.1 for rr.
y=(1+0.1)xy=1.1x\begin{aligned}y &= (1 + 0.1)x\\y &= 1.1x\end{aligned}
Step 2

Determine two points on the line of the equation.

The graph of a direct variation passes through the point (0,0)(0, 0) and (1,k)(1, k), where kk is the constant of proportionality. In this situation, kk is 1.1.
y=kxy=1.1x\begin{aligned} y&=kx\\y&=1.1x\end{aligned}
So, the line of the equation goes through the points (0,0)(0, 0) and (1,1.1)(1, 1.1).
Solution
Plot the points and determine the slope, or kk, of the equation. The graph of the equation goes through the origin with a slope of 1.1.
From the graph, when the price of the item xx is $5, the total cost yy after tax is between $5 and $6. Substitute 5 for xx in the equation to identify the exact total cost.
y=1.1x=1.1(5)=5.5\begin{aligned}y &= 1.1x\\ &=1.1(5)\\&=5.5\end{aligned}
The exact total cost is $5.50.
Step-By-Step Example
Applying Direct Variation to Currency
Currency in U.S. dollars varies directly with the value in other currencies. On a given day, 8 euros are exchanged for 9.25 U.S. dollars. How many euros would be exchanged for 50 U.S. dollars?
Step 1
Write the direct variation equation. Let yy represent the number of euros and xx represent the number of U.S. dollars.
y=kxy=kx
Step 2
Substitute the given values of xx and yy into the equation, and solve for kk.
8=k(9.25)89.25=k0.865=k\begin{aligned}8&=k(9.25)\\ \frac{8}{9.25}&=k\\ 0.865&=k\end{aligned}
This means that the exchange rate is about 0.865 euro per U.S. dollar.
Step 3
Substitute the value of kk into the direct variation equation.
y=0.865xy=0.865x
Step 4
The number of U.S. dollars is given, which is represented by xx. Substitute 50 for the value of xx into the equation.
y=0.865(50)y=43.25\begin{aligned}y&=0.865(50)\\y&=43.25\end{aligned}
Solution
At the given exchange rate, 43.25 euros would be exchanged for 50 U.S. dollars.