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What is algebra?


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Algebra

What is Algebra?

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This topic will cover how to:

  • what algebra is, including definitions of algebraic expressions and equations, constants and variables and like and unlike terms;
  • how to simplify algebraic expressions by adding and subtracting like terms; and
  • how to simplify algebraic expressions by multiplying and dividing.

Algebra

Algebra is a kind of shorthand language that we use to describe patterns and to solve problems.

So just as you need to practice the basics of spelling and sentence structure when you first learn English, you need to practice the basics of algebra so that you have an understanding of the language and can apply it.

Hence you should keep in mind that while solving generic algebraic problems is not something you are likely to do outside of a mathematics classroom, you need to understand these examples so that you can use the concepts in situations that are relevant.

Constants and Variables

When a problem is written in the language of algebra we call it either an algebraic expression (when there is no equals sign), or an algebraic equation (when there is an equals sign).

Both algebraic expressions and algebraic equations involve the use of constants and variables, which are defined as follows:

Constant: Constants are numbers with a fixed or constant value (e.g. 1, 2, 3, ...)

Variable: Variables are represented by small letters of the alphabet, or indeed by any symbol. They are used to represent some unknown number in our expression or equation, and the value of this number can vary (as the word variable suggests) according to the context.

Constant: Constants are numbers with a fixed or constant value (e.g. 1, 2, 3, ...)

Variable: Variables are represented by small letters of the alphabet, or indeed by any symbol. They are used to represent some unknown number in our expression or equation, and the value of this number can vary (as the word variable suggests) according to the context.

Evaluating Algebraic Expressions by Substitution

In the algebraic expression a + 5 the number 5 is a constant and the letter a is a variable because its value varies depending on the context.

Different numbers can be substituted for the variable a and, depending on the number substituted, the value of the algebraic expression a + 5 changes. For example:


When a = a + 5 =
0 0 + 5 = 5
1 1 + 5 = 6
10 10 + 5 = 15
17 17 + 5 = 22

Like Terms

Algebraic expressions can contain more than one variable, for example:

2x + y + 3y + 5x + x

In this case we say that the expression consists of like and unlike terms, which are defined as follows:

Like Terms: Like terms are terms which have the same variable. For example 2x, 5x and x are all like terms in the above expression. Similarly, y and 3y are like terms.

Unlike Terms: Unlike terms are terms which have different variables. For example 2x and y are unlike terms in the above expression.

It is important to understand that only like terms can be added and subtracted- you wouldn't say that 2 apples plus 5 oranges equals 7 apples (or seven oranges), and similarly you can't add 2x and 5y together.

Like Terms: Like terms are terms which have the same variable. For example 2x, 5x and x are all like terms in the above expression. Similarly, y and 3y are like terms.

Unlike Terms: Unlike terms are terms which have different variables. For example 2x and y are unlike terms in the above expression.

It is important to understand that only like terms can be added and subtracted- you wouldn't say that 2 apples plus 5 oranges equals 7 apples (or seven oranges), and similarly you can't add 2x and 5y together.

Simplifying Algebraic Expressions:
Adding or Subtracting Like Terms

As with most things, it is best to write an algebraic expression in the simplest way possible - no need to make it any more confusing than necessary!

To simplify an algebraic expression by adding or subtracting like terms you should follow these steps:

1. Rewrite the expression so that all like terms are next to each other. When you do this, remember to shift the + or - sign that is in front of the term with it.

For example, when simplifying 2a - 3b - a + 4ab- 2b you would rewrite the expression as
2a - a - 3b - 2b + 4ab

2. Simplify by adding or subtracting like terms.

For example, adding and subtracting the like terms in our rearranged algebraic expression 2a - a - 3b - 2b + 4ab gives 2a - a = a and -3b - 2b = -5b, so our simplified expression is a - 5b + 4ab.

1. Rewrite the expression so that all like terms are next to each other. When you do this, remember to shift the + or - sign that is in front of the term with it.

For example, when simplifying 2a - 3b - a + 4ab- 2b you would rewrite the expression as
2a - a - 3b - 2b + 4ab

2. Simplify by adding or subtracting like terms.

For example, adding and subtracting the like terms in our rearranged algebraic expression 2a - a - 3b - 2b + 4ab gives 2a - a = a and -3b - 2b = -5b, so our simplified expression is a - 5b + 4ab.

Simplifying Algebraic Expressions:
Multiplying or Dividing

To simplify an algebraic expression by multiplying or dividing you should follow these steps:

1. Rewrite the expression as a product of its factors.

For example, when simplifying 4ab x 3acd, rewrite it as 4 x a x b x 3 x a x c x d.

2. If necessary, rewrite the expression so that all the like terms are next to each other.

For example, rearranging the factors in our expression so that the like terms are next to each other gives 4 x 3 x a x a x b x c x d.

3. Multiply/divide the constants and multiply/divide the variables, as appropriate.

For example, multiplying and dividing the constants and variables in our rearranged expression 4 x 3 x a x a x b x c x d gives 4 x 3 = 12 and a x a = a2, so our simplified expression is 12a2bcd.

1. Rewrite the expression as a product of its factors.

For example, when simplifying 4ab x 3acd, rewrite it as 4 x a x b x 3 x a x c x d.

2. If necessary, rewrite the expression so that all the like terms are next to each other.

For example, rearranging the factors in our expression so that the like terms are next to each other gives 4 x 3 x a x a x b x c x d.

3. Multiply/divide the constants and multiply/divide the variables, as appropriate.

For example, multiplying and dividing the constants and variables in our rearranged expression 4 x 3 x a x a x b x c x d gives 4 x 3 = 12 and a x a = a2, so our simplified expression is 12a2bcd.

Examples: Simplifying Algebraic Expressions

1. Simplify the following expressions by adding or subtracting like terms:

a.c + 5c - 2c = 4c

b.3d - 5c + 4 - d - 9 + 3c = 3d - d - 5c + 3c + 4 - 9

= 2d - 2c - 5

2. Simplify the following expressions by multiplying or dividing:

a.13a x 2b = 13 x a x 2 x b

= 13 x 2 x a x b

= 26ab

b.10a2b/ 5ac = 10 x a x a x b/5 x a x c

= 210 x a x a x b / 15 x a x c

= 2ab / c

= 4c
= 3d - d - 5c + 3c + 4 - 9
= 2d - 2c - 5
= 13 x a x 2 x b
= 13 x 2 x a x b
= 26ab
= 10 x a x a x b/5 x a x c
= 210 x a x a x b / 15 x a x c
= 2ab / c

Activity 1: Practice Questions

End of Topic

Congratulations, you have completed this topic.

You should now have a better understanding of what algebra is.

 


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