Unit 2: Algebra
Before we dive into Algebra, how about a little background. The branch of mathematics we call Algebra wasn’t always called Algebra. People from all over the world, back to the Babylonians have developed some form of Algebra. But in the early 9th century a Persian Mathematician living in modern day Baghdad, Iraq. What the word Algebra means is to restore or complete something. This is actually not just a trivial piece of information, it provides a lot of insight into the math behind Algebra.
Now you may think that since it is such an old branch of mathematics that it’s fully fleshed out and all is known about it, that nothing is changing. That’s not at all the case. Algebra is such a powerful branch of mathematics because it gives us powerful tools to explore unknown things. As a result the field is not dead, it’s developing. For example, when I took Algebra the Fundamental Theorem of Algebra, which discusses the nature of solutions to polynomial equations, is no longer considered an Algebra topic! Strange, right?
Back to the meaning of the word, Algebra, to restore and make complete. The difference between Algebra and computation is with computation the only thing left to complete a statement is the actual value. For example:
5 + 4 = ____
We don’t usually use something so obvious in Algebra, except to introduce how it is written and some of the ways in which Algebra operates. Suppose we knew the sum, but not one of the numbers in that sum.
5 + ? = 9
This question mark is not normally what we write, but you see it represents the piece of missing information. Algebra means to make complete, so it is essentially asking what number is missing?
? = 4
Now typically we don’t use question marks to represent unknown values (simply called unknowns, which are different from variables, which will be discussed soon). Typically we use a symbol, often a letter of the alphabet. A popular choice is x because when we begin graphing equations, x is traditionally in the input value, the horizontal axis.
So we could write the following with an x:
5 + x = 9
Pro Tip #1: Read the math as a question, not a statement.
As a Question: Five plus what is nine?
As a Statement: Five plus x is nine.
If you have been following my methods for any time now you should understand that I believe in the power of learning how to do difficult problems by examining simple problems. If we form good habits early, the more difficult math will be easier for us.
Algebra seeks to restore the equation by finding the value of the unknown number that would make the equation true…to restore it to its former glory!
Often people get upset with variables and unknowns. They complain, “Math was fine until the alphabet was introduced.”
There are two responses I have to this complaint. First, the symbols in the alphabet are just symbols. We can use them to mean many different things. Can you imagine a pirate movie where they were using a map and their adventure went perfect until they were confused by the letter x on the map? Absurd, right?
We use letters to represent other things that are not at all letters, too. Isles in grocery stores are often assigned a letter name. Groups of workers assigned to a task will be assigned letter names, as well.
The second comment is this: In Algebra the unknowns will be numbers. We may not know exactly which number the unknown or variable will be, but we know a lot about numbers. We will also be given clues as to what the number is. Algebra uses the same mathematical operations you know and love, before the alphabet was introduced.
I really believe that the complaint about the alphabet being added to math derailing everything is just something to latch onto and take the blame for weak mathematical foundations. That is why so much time was spent on the number unit, to shore up any weak foundations.
Consider the equation:
3x + 1 = 10
If you read the equation like a question, I bet you (regardless of mathematical experience) can answer the question. In other words, I bet by reading the equation as a question you can perform the Algebra (restore the equation).
Three times what plus one is ten?
Three x plus one is ten.
Again, this is not a complicated problem, but by asking the question instead of just reading it as a statement we are aligning our thinking with the purpose of Algebra.
Not convinced yet?
The square root of, 25 plus what number, plus the square root of seven times that same number equals five?
The square root of 25 plus x, plus the square root of 7x is 5.
One important thing to notice here. The statement loses the operation of multiplication. “seven x,” is not really an operation.
Did you figure it out yet? What value of x, what number, would make this true?
If you don’t have it yet, give yourself a second. Read it as a question, again. Look for relationships between the different terms. (Terms are separated, here, by addition and the equal sign.)
But, do not try to use any formal methods of solving this equation. It’s really a simple answer, once you see what’s being asked.
The value for x that restores this equation is zero.
Variables are different than unknowns. An unknown has one, or maybe two values. Variables are changing, having a large number of possible values.
In our first example, 5 + x = 9, there is only one value of x that would restore that equation (solve it). But in the equation below, there are many values.
3x + 2y = 6
Because we have two variables, an x and a y, we have two variables, they are actually related to one another by a constant of variation, which sounds like a fantastic oxymoron. A great oxymoron is jumbo shrimp, or perhaps you like the oxymoron presented in the fact that we park in a driveway and drive on a parkway.
Regardless, constant of variation means that there’s a constant, unchanging ratio that describes how the variables x and y are related. It’s also referred to as slope, maybe you’ve heard of that.
Either way, we now have variables because there are infinitely many values of x and y that could solve this equation. However, for this equation each x will have only one value of y. That’s why we would write these as ordered pairs. Ordered pairs are alphabetically in order (where ordered comes from), and pairs means of course two.
By reading the algebraic equation 3x + 2y = 6, can you come up with a solution?
If not, give yourself a moment. Develop a sense for how these things work. Don’t be frustrated, if you don’t see an answer yet, that’s quite alright.
There are infinitely many answers. We could never write them all. One possible answer is x = 2 and y = 0. As an ordered pair that would be written as (2, 0). Another answer is if x = 0 and y = 3. That would be written (0, 3). Because x comes before y in the alphabet, we write the x value first and the y value second.
- Algebra means to restore (by solving) an equation
- The solution is a value that makes the equation true
- Reading algebraic equations like a question is powerful because all you need to do is answer the question to solve the equation
- Algebra uses the math you already know, covered in the order of operations (PEMDAS)
- An unknown has one value
- A variable has many values
Practice Problems. Write the question the algebraic equations are really asking:
- 3x + 4 = 1
- 8 – x = 0
- 5x + 2 = 3x + 2
- 2x = 8
For additional practice, without performing any inverse operations, what are the solutions to these four problems?
There are problems posted in the Practice Problems icon at the top.