Real World Applications

Many real-world applications can be modeled by linear equations. Things like cell phone packages that charge a monthly service fee plus a charge per minute of talk time, and a car rental company's daily fee plus amount per miles driven.

Modeling Real World Equations

To set up or model a linear equation to fit a real-world application, we must first determine the known quantities (constants) and define the unknown quantities (variables). After than we can interpret the words as mathematical expressions using mathematical symbols.

$\underline{Example}$

Let's imagine that the cell phone company has a plan that charges $\$15$ every month plus $\$0.02$ per minute of talk time.

The known quantities are the $\$15$ monthly charge and the $\$0.02/min$ charge. However, the unknown quantity is the number of minutes of talk time.

In this case the $\$0.02$ is multiplied by the unknown quantity so we can express this as $0.02x$. The $\$15$ is a constant charge every month. The verbal statement says plus so we are adding the charges together.

Assuming $y$ is the cost of the bill, this bill can be modeled as $y = 0.02x + 15$

Finally, some verbal expressions we can directly translate into math...

Verbal	Mathematical Translation
One number exceeds another by $a$	$x$, $x + a$
Twice a number	$2x$
One number is $a$ more than another number	$x$, $x + a$
One number is $a$ less than twice another number	$x$, $2x - a$
The product of a number and $a$, decreased by $b$	$ax - b$
The quotient of a number and the number plus $a$ is three times the number	$\frac{x}{x + a} = 3x$
The product of three times a number and the number decreased by $b$ is $c$	$3x(x - b) = c$

Common Formulas

Some common formulas we encounter solving real world problems include...

1. Area of a rectangular region: $A = LW$ where $L$ is the length and $W$ is the width.
2. Perimeter of a rectangle: $P = 2L + 2W$ where $L$ is the length and $W$ is the width.
3. Volume of a rectangular solid: $V = LWH$ where $L$ is the length, $W$ is the width, and $H$ is the height.