Introduction to Variables

Introduction

In this lesson, you will learn about variables and practice multiplication equations involving variables.

Just for Fun

Why is X often used as a variable?

On June 15, 2015, Terry Moore wrote in Cosmos Magazine that much of what is known about mathematics originated in Persia and Arabia. The word algebra literally comes from the Arabic word al-jabr which means “the reunion of broken parts.” When Spanish scholars were translating Arabic mathematical texts, they often came across the Arabic word shay-un which means “something.” Since Spanish doesn’t have the “sh” sound, they used the Greek letter chi (X) instead. Later, these Spanish texts were translated into Latin, X became the standard symbol for something unknown.

Gospel Connection

The example of X becoming the standard variable in algebra is an example of how texts can change over time through various translations. The 8th Article of Faith says that “we believe the Bible to be the word of God as far as it is translated correctly.” The Lord revealed to Joseph Smith in the Joseph Smith Translation of the Bible many clarifications for scriptures in the Bible that had lost their original meaning due to changes or mistakes in translation. Some of this can be found in Joseph Smith—Matthew in the Pearl of Great Price.

These videos illustrate the lesson material below. Watching the videos is optional.

Introduction to Variables

A variable represents a number that can change or that you don’t know yet. Because you don’t know the number, you use a letter to represent it. Most often you use x and y, but any letter or symbol can be used as a variable.

Example 1
Imagine that you sell necklaces for a living. You may work hard to sell them, but you won’t sell the same number of necklaces each day. One day you may only sell two or three necklaces, but another day you may sell 10 or 20.

The number of necklaces sold is a variable because it changes and is a number that is outside of your control. However, you can control how much the necklaces sell for, and you can write an equation that helps determine how much you will make. If each necklace is sold for $2, your equation would look like this: $$2\times n = Total\;Cost$ where $n$ represents how many necklaces will be purchased.

Read aloud, the equation would be something like “two dollars times the number of necklaces sold equals how much money you will make.”

If you sold 3 necklaces ($n=3$), you could calculate $$2\times3=$6$ and know that you made $6 that day.
If you sold 10 necklaces ($n = 10$), you could perform the same equation $$2\times10=$20$ to find out that you made $20.

In this example, you can use a picture or anything to represent the variable. In algebra, letters are usually used to represent the variables, so you can use n to represent the “necklace” as shown above. One of the most common variables is the letter x.

Different Multiplication Symbols

One multiplication symbol looks like a lowercase x, so when you have an expression or an equation with variables, you'll want to use something besides that multiplication symbol to indicate multiplication.

Here are three different ways to represent multiplication in algebraic expressions or equations:

A dot
- This dot is not a decimal point. It is placed up higher, about halfway between the numbers being multiplied together: $3\cdot2 = 6$.
Parentheses
- When one number is next to another number that is contained within parentheses, it means that you need to multiply the two numbers together: $3(2) = 6$.
Variables
- When a number is right next to a variable, like n or x, it means that you need to multiply the number by the variable: $3n$.

Multiplication Written in Different Ways

Example 2
Simplify this expression: $3\cdot4 - 5(2)$

As you follow the order of operations, you will perform the two multiplication operations and then perform the subtraction operation:
\begin{align*}
&3\cdot4 - 5(2) &\color{red}\small\text{Simplify using order of operations}\\\\
&12 - 10 &\color{red}\small\text{Multiply or divide from left to right}\\\\
&2 &\color{red}\small\text{Add or subtract from left to right}\\\\
\end{align*}

The correct answer is 2.

Example 3
Simplify this expression: $9n\cdot5$

9n indicates that 9 is being multiplied by n. There are several ways that you could rewrite it, and all of them are appropriate so long as everything continues to be multiplied together.

\begin {align*}9n\cdot5\\9\cdot n\cdot5\\9\cdot5\cdot n\\5\cdot9\cdot n\\\end{align*}

Solve this problem as much as you can, but because there is a variable in the expression, you will not be able to solve it completely.

\begin{align*} &9n\cdot5 &\color{red}\small\text{Simplify using order of operations}\\\\ &9\cdot5\cdot n &\color{red}\small\text{Rearrange the order}\\\\ &45n &\color{red}\small\text{Multiplication property}\\\\ \end{align*}

Example 4
Simplify this expression: $9n\cdot 5p$

This expression has two variables. Since everything is being multiplied together, you can move elements around in this equation several ways: $9n\cdot 5p = 9\cdot5np = 45np$. Ultimately, you will get $45np$ as the correct answer. As a note, when multiple variables are in an equation, you normally alphabetize them if they remain in the answer.

Example 5
Simplify this expression: $4(5 - 6) + 3^2\cdot 5$

In this algebraic expression, multiplication is indicated with both the parentheses and a dot. We need to keep that in mind as we simplify the expression.

\begin{align*}4(5 - 6) &+ 3^2\cdot 5&\color{red}\small\text{Simplify using order of operations} \\\\4(-1) &+3^2\cdot 5 &\color{red}\small\text{Solve within the parentheses first}\\\\ 4(-1) &+9\cdot 5 &\color{red}\small\text{Solve the exponents}\\\\-4 &+ 45 &\color{red}\small\text{Multiply or divide from left to right}\\\\& 41&\color{red}\small\text{Add or subtract from left to right} \\ \end{align*}

The final answer is 41.

Things to Remember

A variable is something that can change or differ.
A dot, parentheses, or a variable can all be used to indicate multiplication in algebraic equations.
Keep parentheses around a negative number so the negative sign isn’t misinterpreted as a minus sign.

Practice Problems

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