Constants, Variables, and Algebraic Expressions

It’s almost back-to-school time! Perish the thought, I know . . . at our house, we will be in denial until the big day, if not longer. The kids at school will all be starting off in their math classes with some basic terms and definitions, so I thought I’d get us off to a start with some basic algebra.

The words “algebraic expression” may sound really complicated, but really it is just a group of numbers and letters, (constants and variables) that represent (express!) something. Everything in an algebraic expression is either a constant or a variable. The constants always remain the same (thus the term constant). They are numbers, like the 2 and 5 in 2x + 5, or the price of a loaf of bread at the grocery store, and their values obviously don’t change (unless there is a sale!)

Variables are represented by letters in an expression, and they can change. In our algebraic example above (2x + 5), the variable is x. In a real-life situation, a variable might be the number of loaves of bread you buy at the store. It can change with every trip to the store.

Now, the magic of algebra is that you can write one equation, but use it for a number of different scenarios. In our grocery example, you might say that one week you bought 2 loaves of bread. If bread costs $2.19/loaf, you will have spent 2 x $2.19, which equals $4.38. But what if you had bought 3 loaves? Or only 1? That’s where algebra comes in handy. Instead of writing 2 x $2.19, 4 x $2.19, or 1 x $2.19, let’s just write the “ x $2.19” part once, recognizing that no matter how many loaves we buy, we will multiply that number by $2.19.

______ x $2.19 = the amount you spent on bread

Now, let’s use the variable x for the number of loaves you bought. The only thing is, if I write x times $2.19, it will look like this: x x $2.19. That’s kind of confusing, isn’t it? So to keep us from confusing x and times, there are some new ways to represent multiplication:

1)instead of an x, in algebra, we use a dot between the two factors. I don’t know how to type this, but imagine the following without the bottom dot of the colon
4 : 3 = 12
2)we also use parenthesis to mean the same thing:
(4)(3) = 12
3)finally, whenever there is a variable and a constant (a letter and a number), we just write them together whenever we mean multiplication:
4x = 12

Ok, back to our example. Using an x for the number of loaves you bought, we have:

(x)(2.19) = the amount you spent on bread, or
2.19x = the amount you spent on bread.

Now you just use whatever number you want for x. It could be a 2, or a 10. You do the problem the same way!

Yeah, it’s basic, but that is the magic of algebra!!