Introduction

Welcome to a basic introduction of algebra. In this tutorial, we will review some of the more common algebraic concepts.

Definitions

To start with, let's define some of the more common terms used in algebra:

Examples

Sometimes it is easier to understand the definitions when you have a physical example to look at. Here is an example of the above terms.

x + 5 = 12

In this above example, we have:

Solving Equations

The primary use for algebra is to determine an unknown value, the "variable", with the information provided. Continuing to use our example from above, we can find the value of the variable "x".

x + 5 = 12

In an equation, both sides result in the same value. So you can manipulate the two expressions however you need, as long as you perform the same operation (or change) to each side. You do this because the goal when solving an equation is to get the variable into its own expression, or by itself on one side of the = sign.
For this example, we want to remove the "+5" so the "x" is alone. To do this, we can subtract 5, because subtraction is the opposite operation to addition. But remember, we have to perform the same operation to both sides of the equation. Now our equation looks like this.

x + 5 - 5 = 12 - 5

The equation looks like a mess right now, because we haven't completed the operations. We can simplify this equation to make it easier to read by performing the operations "5-5" and "12-5". The result is:

x = 7

We now have our solution to this equation!

Solving Equations II

Let us look at a slightly more challenging equation.

3x + 4 = 13

Again we can start with subtraction. In this case, we want to subtract 4 from each side of the equation. We will also go ahead and simplify with each step. So now we have:

3x = 9

"3x" translates to "3*x", where the "*" symbol indicates multiplication. We use the "*" to avoid confusion, as the "x" is now a variable instead of a multiplication symbol. The opposite operation for multiplication is division, so we need to divide each expression by 3.

x = 3

And now we have our solution!

Solving Equations III

Now we are getting in to more complex operations. Here is another equation for us to look at:

x^2 - 8 = 8

Our very first step will be to add 8 to each side. This is different from our previous examples, where we had to subtract. But remember, our goal is to get the variable alone by performing opposite operations.

x^2 = 16

But what does the "^2" mean? The "^" symbol is used to denote exponents in situations where superscript is not available. When superscript is available, you would see it as x2. For the sake of this project, however, we will use the "^" symbol.
An exponent tells you how many times the base (in our case, "x") is multiplied by itself. So, "x^2" would be the same as "x*x". Now the opposite function of multiplication is division, but we would have to divide both sides by "x". We do not want to do this, as that would put an "x" on the other side of the equation. So instead, we need to use the root operation! For an exponent of "2", we call this the "square root" and denote it with "√". Our equation is now:

x = √9

Performing a root operation by hand can be a tedious process, so we recommend using a calculator when necessary. However, we are lucky in that "9" is a perfect square, so we do not need to calculate anything. Instead, we find our answer to be:

x = 3

System of Equations

As you explore your algebra studies further, you may start to run across equations with more than one variable. The first such equations will likely look like:

y = 3x

An equation like this does not have one single solution. Rather, there are a series of values for which the equation is true. For example, if "x=3" and "y=9", the equation is true. These equations are usually used to plot a graph.
Getting more complicated, though, you may be given a pair of equations. This is called a "system of equations", and CAN be solved. Let's look at how we do this! Consider the following system of equations:

y = 3x | y - 6 = x A system of equations IS solvable, but it is a multi-step process. To get started, we need to chose a variable we are solving for. Let's solve for "x" first. From the second equation, we know that "x" equals "y - 6", but we cannot simplify that further because we do not have a value for "y". Except, thanks to the system of equations, we DO have a value for "y". We know that "y" equals "3x". So, looking at our second equation, we can replace "y" with "3x" because they have the same value. We then get:

3x - 6 = x

Now we can solve for "x"! We start by adding 6 to each side.

3x = x + 6

We still need to get "x" by itself, so we subtract "x" from both sides and get:

2x = 6

If this confuses you, remember that "3x" is the same as "x+x+x". Subtract an "x" from that and you get "x+x", or "2x". Now we divide both sides by 2 and have our value for x!

x = 3

However, our work is not done yet. We still need to find the value for "y". Let's go back to our first equation:

y = 3x

We have a value for "x" now, so let's see what happens if we put that value in.

y = 3*3

We perform the multiplication and discover that "y=9"! Our solution to this system of equations then is:

x = 3 and y = 9

Try it Yourself!

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