This section includes:
Introduction to Algebra
Can you really mix numbers and letters? Of course! Let’s begin to learn algebra!
Now 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 say that you landed a new job working at the mall at your favorite clothes store and you make $10 an hour – yeah! Of course how much you make depends on how many hours you work. So 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. So 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.
So 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
Let’s turn 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. 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.
So let’s fill in some numbers for \(n\), to see how much you will make for the example above:
Number of hours you work that week (independent variable) |
Earnings you make that week (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”.
So 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; these are described in 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.
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On to Types of Numbers and Algebraic Properties – you are ready!