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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

Expert Contributor
Will Welch

Will has a doctorate in chemistry from the University of Wyoming and has experience in a broad selection of chemical disciplines and college-level teaching.

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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Additional Activities

Writing Equations for Everyday Life

Symbols and variables in algebra often confuse people, but they generally represent normal quantities and we subconsciously do algebra all the time.

In the following exercises, take everyday situations and write corresponding algebraic expressions and equations (do not solve).

Combining like terms:

  • My dad brought home 2 apples, 5 bananas and an orange and put them in the fruit bowl. My mom came home with 1 apple and 3 oranges and put them in the bowl. Write expressions for what dad brought home and what mom brought home. Use a for apple, b for banana, and o for orange. Then add the two expressions and write an expression for what is finally in the bowl.

Multiplying variables and sums:

  1. Everyone in the class gets 3 erasers at the beginning of the school year. Use n for number of students and e to designate erasers and write an expression for how many erasers are given out.
  2. Three students were added to the class the next day and they each got 3 erasers too. Write an expression for the new number of erasers given out.

Division:

  • A pizza having 16 slices is to be split between 5 people. Let s be slices and p be people and write an expression for how many slices each person gets.

Equations:

  1. I paid $5 for a loaf of bread and got $2.60 back. Write an equation including a variable for the price of the bread.
  2. Admission to a concert costs $20 and parking costs $10 per car. Let p be the number of people in one car and c be the total cost of them going to the concert. Write an expression for the cost of them attending the concert.

Answers

Combining like terms:

  • Dad: 2a + 5b + o
  • Mom: a + 3o
  • Total: 2a + 5b + o + a + 3o = 3a + 5b + 4o

Multiplying variables and sums:

  1. 3n
  2. 3(n + 3)

Division:

  • 16s / 5p

Equations:

  1. x = $5 - $2.60 or $2.60 = $5 - x or $5 = x + 2.60
  2. c = $20p + $10

Follow-up

Look back at the expressions and equations and simplify or solve where possible. Then look at ones that cannot be solved. Discuss why it is important and helpful to have variables for unknown quantities.

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