Objective 1: Introduce Variables and Formulas

A small business produces quality pens imprinted with your custom logo. They charge a set-up fee of $120 plus $2 for every pen ordered. Calculate the cost of particular orders as follows: \[ \begin{array}{c}\$ 2\left( { \color{blue}{50} } \right) + \$ 120 = \$ 220\,\,\,\,\,\,\,\,\,\, \leftarrow {\rm{ 50 \,custom \,pens}}\\\$ 2\left( \color{blue}{100} \right) + \$ 120 = \$ 320\,\,\,\,\,\,\,\,\,\, \leftarrow {\rm{ 100 \,custom \,pens}}\\\$ 2\left( \color{blue}{300} \right) + \$ 120 = \$ 720\,\,\,\,\,\,\,\,\,\, \leftarrow {\rm{ 300 \,custom \,pens}}\end{array} \] Notice the pattern in the calculations. The number of custom pens ordered is the only number that changes. In algebra, letters are used to represent numbers that can change and are called variables. In this case, if we let the letter \(n\) represent the number of custom pens ordered then we can generalize this calculation as follows: \[2\color{blue}{n} + 120\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm{Cost \,of \,n \,custom \,pens}}{\rm{.}}\] That is, the cost is $2 times the number of custom pens ordered plus the $120 set-up fee. Using variables to generalize this calculation marks the beginning of our transition from arithmetic to algebra. Doing this leads to reusable formulas, which are mathematical models used to describe common applications. Now using the compact formula, we can efficiently calculate the cost of a 1,000-pen order. \[\begin{array}{c}Cost = 2n + 120\\ = 2\left( \color{blue}{1,000} \right) + 120\\ = 2,000 + 120\\ = 2,120\end{array}\] Here 1,000 pens would cost $2,120. Formulas such as this, used in conjunction with computers and machines, are responsible for the increased productivity we see in most all areas of everyday life. Soon you will begin to see algebra everywhere!

Example 1: Identify the variable in the given formula. \[{\rm{Degrees \,Celcius }} = \frac{{5\left( {F - 32} \right)}}{9}\] Solution: The variables are the letters that represent numbers. In this case, there is only one variable \(F\).
Answer: \(F\)

Try this! Identify the variables: \({a^2} + 5ab\)
Answer: \(a\) and \(b\)

Two common geometric figures are the rectangle and square. Recall that the perimeter is defined to be the sum the lengths of the outside edges. The area, on the other hand, is the amount of space inside the figure measured in square units. There are nice formulas for the area \(A\) and perimeter \(P\). For the rectangle, the variable \(l\) represents the length of the rectangle and \(w\) represents the width.

\[\begin{array}{l}P = 2l + 2w\,\,\,\,\,\,\,\,{\rm{Perimeter \,of \,a \,rectangle}}{\rm{.}}\\A = l \cdot w\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm{Area \,of \,a \,rectangle}}{\rm{.}}\end{array}\] For the square, the variable \(s\) represents the lengths of all four equal sides.

\[\begin{array}{l}P = 4s\,\,\,\,\,\,\,\,\,\,\,{\rm{Perimeter \,of \,a \,square}}{\rm{.}}\\A = {s^2}\,\,\,\,\,\,\,\,\,\,\,{\rm{Area \,of \,a \,square}}{\rm{.}}\end{array}\] There are many more formulas associated with various geometric figures. You can view some of them now by looking into appendix a.

Exercises: 
Identify the variable(s).

1.  \(7{x^3}\)
2. \(5{y^8}\)
3. \(2x + 4y\)
4. \(5x - 3y\)
5. \(7{a^2} - 5a + 2\)
6. \(\frac{{3b - 10}}{{{b^2} + 3}}\)