### Variables

No one likes doing the same problem over and over again, that’s why mathematicians invented algebra. Algebra takes the basic principles of math and makes them more general, so a problem can be solved once and then that solution used to solve a group of similar problems.

In arithmetic, you’ve dealt with numbers and their arithmetical operations (such as \begin{align*}+, \ -, \ \times, \ \div\end{align*}). In algebra, we often use symbols called **variables** (which are usually letters, such as \begin{align*}x, \ y, \ a, \ b, \ c, \ \ldots\end{align*}) to represent numbers, and sometimes processes.

For example, we might use the letter \begin{align*}x\end{align*} to represent some number we don’t know yet, but need to identify in the course of a problem. Or we might use two letters, like \begin{align*}x\end{align*} and \begin{align*}y,\end{align*} to show a relationship between two numbers without needing to know what the actual numbers are. The same letters can represent a wide range of possible numbers, and the same letter may represent completely different numbers when used in two different problems.

Using variables offers advantages over solving each problem “from scratch.” With variables, we can:

- Formulate arithmetical laws such as \begin{align*}a + b = b + a \end{align*} that apply to all real numbers in place of \begin{align*}a\end{align*} and \begin{align*}b\end{align*}.
- Refer to “unknown” numbers. For instance: find a number \begin{align*}x\end{align*} such that \begin{align*}3x + 1 = 10\end{align*}.
- Write more compactly about functional relationships such as, “If you sell \begin{align*}x\end{align*} tickets, then your profit will be \begin{align*}3x - 10\end{align*} dollars, or \begin{align*}f(x) = 3x - 10\end{align*}, where \begin{align*}f\end{align*} is the profit function, and \begin{align*}x\end{align*} is the input (how many tickets you sell).

#### Writing an Algebraic Equation

Write an algebraic equation for the perimeter and area of the rectangle below.

To find the perimeter, we add the lengths of all 4 sides. We can still do this even if we don’t know the side lengths in numbers, because we can use variables like \begin{align*}l\end{align*} and \begin{align*}w\end{align*} to represent the unknown length and width. If we start at the top left and work clockwise, and if we use the letter \begin{align*}P\end{align*} to represent the perimeter, then we can say:

\begin{align*} P = l + w + l + w\end{align*}

Since we are adding two \begin{align*}l's\end{align*} and two \begin{align*}w's\end{align*}, we can say that:

\begin{align*}P = 2 \cdot l + 2 \cdot w\end{align*}

It's customary in algebra to omit multiplication symbols whenever possible. For example, \begin{align*}11x\end{align*} means the same thing as \begin{align*}11 \cdot x\end{align*} or \begin{align*}11 \times x\end{align*}. We can therefore also write:

\begin{align*} P = 2l + 2w\end{align*}

Area is length multiplied by width. In algebraic terms we get:

\begin{align*} A = l \times w \to A = l \cdot w \to A = lw\end{align*}

Note: \begin{align*}2l + 2w\end{align*} by itself is an example of a **variable expression**, whereas \begin{align*} P = 2l + 2w \end{align*} is an example of an **equation**. The difference between expressions and equations is the presence of an equals sign (=).

In the above example, we found the simplest possible ways to express the perimeter and area of a rectangle when we don’t yet know what its length and width actually are. Now, when we encounter a rectangle whose dimensions we do know, we can simply **substitute** (or "plug in") those values in the above equations. In Algebra, you will encounter many expressions that can be evaluated by substituting values for the given variables.

#### Writing Equations Involving Variables

1. Eric has some money in his savings account. How much more money does he need in order to buy a game that costs $98?

Let \begin{align*}M\end{align*} represent the money that Eric still needs, and let \begin{align*}S\end{align*} be the money that he has in his savings account. Then, by subtracting the money he already has from the total cost of the game, we can figure out how much money he still needs:

\begin{align*}M=98-S\end{align*}

2. Write an equation for the sum of 3 times some number and 5.

Let \begin{align*}S\end{align*} be the total sum. Let \begin{align*}n\end{align*} be "*some number."* Then "*3 times some number"* is \begin{align*} 3\cdot n\end{align*} and then the sum of \begin{align*}3 \cdot n\end{align*} and 5 is:

\begin{align*}S=3n+5\end{align*}

Watch this video for help with the Examples above.

### Example

#### Example 1

Alex has a certain amount of nickels and dimes in a jar. Write an algebraic equation for how much money she has, in terms of the number of nickels and dimes.

Let \begin{align*}n\end{align*} be the number of nickels, and \begin{align*}d\end{align*} be the number of dimes, that Alex has in the jar. Since each nickel is worth $0.05, the amount of money she has in nickels will be \begin{align*}0.05 \text{ times }n, \text{ or }0.05n.\end{align*}

Since each dime is worth $0.10, the amount of money she has in dimes will be \begin{align*}0.10d.\end{align*}

This means that the total amount of money, \begin{align*}M\end{align*}, that Alex has is \begin{align*}M=0.05n+0.10d.\end{align*}

\begin{align*}M=0.05n + 0.10d\end{align*}

### Review

For 1-4, write the following in a more condensed form by leaving out the multiplication symbol.

- \begin{align*}2 \times y\end{align*}
- \begin{align*}1.35 \cdot y\end{align*}
- \begin{align*}3 \frac { 1 } { 4 } \times m\end{align*}
- \begin{align*}\frac { 1 } { 4 } \cdot z \end{align*}

For 5-10, write an equation for the following situations.

- The amount of money Andrea has in a jar full of quarters and dimes.
- The amount of money Michelle has in her coin purse if it only contains quarters, dimes and pennies.
- The sum of 7 and 6 times some number.
- 4 less than 20 times some number.
- The amount of money you will earn if you are paid $10.25 an hour and spend $4.00 round trip to get to and from work.
- A father earns a $2000 dividend from an oil investment and distributes it equally amongst his children.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 1.1.