Introduction to Algebra | More Practice |

## Introduction to Algebra

Can you really mix numbers and letters? Of course! Let’s begin to learn algebra! This is the beginning of the math that looks difficult, but a lot of the math is actually **easier **than what you’ve been doing, especially compared to fractions. Let’s start with an example.

Let’s say that you landed a new job working at the mall at your favorite clothes store and you make **$10** an hour. Of course, **how much** you make depends on **how many hours** you work. Thus, you need to know how many hours you worked to know how much you make each week.

We can express how much you make by a mathematical expression. A mathematical** expression **is any mathematical term that contains numbers and/or letters (**variables**) and other symbols, like a “$ +$” or a “$ –$ ”. It could be a sum, a difference, a product, or a quotient, or any combination. A mathematical **equation** is simply at least two mathematical expressions with an equal sign between them. Think of a mathematical expression like a phrase in English, and a mathematical equation like a sentence in English.

A **variable** is a letter that represents a number — that’s it! This is what **algebra** is all about!

You don’t know it, but you are using algebra in everyday life. When you go to pay for something, there is an algebraic equation in the computer that figures out the tax you owe. When you go to the bank, there are equations in the computers there that add and subtract the money you have. And so on, and so on.

Getting back to our mall job example, we can express the number of dollars you make each week as the following mathematical equation:

** Weekly Earnings **$ =\$10\times n$, where $ n=$ the* *number of hours you work each week. Turning this into math: $ \boldsymbol{E=10n}$** **(this means

**10**

**times**$ n$).

We need the variable “$ n$” since every week it **varies** as to how many hours you work; thus the word **variable**. The number of hours, or “$ n$” is called the **independent variable**, since it doesn’t depend on anything else (except maybe how nice your boss is!). The variable “$ E$” is called the **dependent variable**, since it depends on the variable “$ n$” (the more you work, the more you make). If you’re confused about which variable is independent and which is dependent, usually the **dependent variable** is what the math **question is asking for**, like “earnings” above.

The number before the variable (**10** in this case) is called the **coefficient** of that variable. If we had $ \displaystyle \frac{n}{{10}}$, the coefficient would be $ \displaystyle \frac{1}{{10}}$, since that would be the same as $ \displaystyle \frac{1}{{10}}n$. Any number by itself in an expression (for example, if we had $ \displaystyle \frac{n}{{10}}+3$) is called a **constant**; in this case the **3** would be the constant.

Let’s fill in some numbers for $ n$, to see how much you will make for the example above:

(independent variable) |
(dependent variable) |

$ n=4$ | $ E=\$10\times 4=\$40$ |

$ n=10$ | $ E=\$10\times 10=\$100$ |

$ n=20$ | $ E=\$10\times 20=\$200$ |

Not too bad, right? You’re doing algebra!

We usually indicate the independent variable by $ \boldsymbol{x}$, and the dependent variably by $ \boldsymbol{y}$ (don’t ask me why). Notice that when the $ x$’s are all spaced evenly apart (**1** apart), the $ y$’s all go up by **10**; in other words, the differences between the $ y$*’*s are **10**. (and We’ll see later that **10** is the **slope** or **rate**, since the $ x$’s are all going up by **1**.

Sometimes we need to work **backwards**: we know the earnings we made that week, but want to figure out how many hours we worked. This is why algebra is so extremely helpful — because a lot of times it’s very difficult to work backwards “in our head”.

As an example, let’s say we made **$100** last week and someone asked us how many hours we had to work to make that much. You can easily see that we worked **10** hours, since $ 10\times \$10=\$100$. But sometimes it’s not that easy.

The way we do it in algebra (at least most of algebra) is to **solve **for the variable, or put it **by itself on one side** of the equation. **This is what algebra is all about! **We can do this by adding, subtracting, multiplying, or dividing the **same number** to both sides, since we are balancing the equation.

It’s like if you and your friend always have the same number of shoes. If one of you gets one more pair (or gives away a pair), the other one must too – to have the same number of shoes for both of you, you need to add or subtract the same number.

To be able to solve for algebraic variables “legally”, we must use what we call **Algebraic Properties** from the **Types of Numbers and Algebraic Properties **section.

And remember again that your goal in solving these algebra problems is **to get the variable or unknown to one side all by itself**! We’ll see this in the **Solving Algebraic Equations** section.

**Learn these rules, and practice, practice, practice!**

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the **Mathway** site, where you can register for the **full version** (steps included) of the software. You can even get math worksheets.

You can also go to the **Mathway** site here, where you can register, or just use the software for free without the detailed solutions. There is even a Mathway App for your mobile device. Enjoy!

On to **Types of Numbers and Algebraic Properties **– you are ready!