Basic Algebra: Rules, Equations & Examples

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  • 0:03 What Is Basic Algebra?
  • 0:57 Adding or Subtracting…
  • 2:38 Multiplying and Dividing Terms
  • 5:17 Solving for the Variable
  • 6:36 Lesson Summary
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Lesson Transcript
Instructor: David Karsner

David holds a Master of Arts in Education

Algebra is the foundation for all higher levels of math. In order to understand any mathematical field, you need to be able to speak algebra, which involves things like solving an equation, isolating the variable, and simplifying the expression.

What Is Basic Algebra?

Basic algebra is the field of mathematics that it one step more abstract than arithmetic. Remember that arithmetic is the manipulation of numbers through basic math functions. Algebra introduces a variable, which stands for an unknown number or can be substituted for an entire group of numbers. Arithmetic poses questions like 2 + 5 = ? Algebra, on the other hand, asks questions like: If x + 5 = 7, what is the value of x? Instead of immediately finding a basic sum, we have to do additional work to solve for an unknown.

Algebra is also a gateway field of mathematics. Once you master the basics, you'll have the tools to talk about higher levels of mathematics and the background understanding to build on as you progress. This lesson contains examples of the most commonly called upon tasks to be performed in algebra.

Adding or Subtracting Like Terms

To add or subtract any terms in algebra, your terms must be like terms, which have the same variable and are raised to the same power. If you have like terms, you add or subtract the numbers attached to the variable, called the coefficients. The variable itself is unchanged. Let's look at an example:

-3x+4y+5x-3y example

In the first step, as you can see, we rearranged our terms to group them by like terms. After that, we rewrote the like terms to have just the coefficients inside parenthesis and the variable outside. This is not necessary, but it helps to show the math we completed in our final step to arrive at the answer of 2x + y.

This type of math is called simplifying expressions. When you look at an algebraic expression like this, it's important to think about the + and - signs as attached to a term. Consider the following:

equivalent expressions

Any time we subtract a number from another, we can rearrange that as we've just done. The number being subtracted becomes a negative number, and the number we are subtracting from is added to the negative number. As long as you keep the signs attached to their terms, you don't change the value of your equation.

Let's consider one more example:

matching like terms

Look carefully at this example. We have three different variables to think about: n, x, and nx. We can't combine any of these three together, so we are left with three terms at the end. Can you see how we applied the same steps as earlier to come to this answer?

Multiplying and Dividing Terms

To multiply or divide terms, you do not have to have like terms. This is different from addition and subtraction, so be careful! Take a look at the following example of basic multiplication with a variable:

multiplication in algebra

We must multiply every part of the equation inside parentheses by 2, and this example shows that happening in the second step, where it's gone from (x + 5) * 2 to (2 * x) + (2 * 5). After we multiply, our final answer is a simple 2x + 10.

Now that we can do the basics, let's make it a little harder and include another variable:

double variable multiplication

Despite having two variables this time, our procedure is the same as it was in the first example, in which (x + 3) * y turns into (y * x) + (y * 3). In other words, we have to multiply everything inside the parentheses by our variable y, which leaves us with an answer of xy + 3y. We can't combine our xy with y, so this is the simplified expression.

The opposite of multiplication is division. There are several main ways to express division. The one that is probably familiar to you uses a symbol that looks like this '÷' and you place it between two numbers. For example: 4 ÷ 2 = 2. We can express this without the symbol, though, by creating a fraction that represents the same thing.

division format

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