Simplifying algebraic expressions

In a nutshell

Algebra uses a combination of letters and numbers to derive expressions called terms. It helps to find unknown quantities. If there is an unknown quantity, start by giving it a letter name, e.g. $$x$$. This can then be used in an expression, equation or formula. 

Terms

Terms are a combination of numbers and letters. Algebraic terms which have the same letter are called 'like terms' and can be added or subtracted, e.g. $$2x +3x=5x$$. This is called simplifying. If terms have different letters, or a different combination of letters, they cannot be added or subtracted, e.g. $$2x+3y$$ cannot be simplified further.

Note: Take care with negatives. The subtract sign belongs to the term it sits in front of. E.g. In $$2x -5y$$, the subtract sign belongs to $$5y$$.

Examples

​$$\begin{aligned}2\times x&=2x\\ \ \\8\times t&=8t\\ \ \\-5\times y&=-5y\\ \ \\3\times n\times n&=3n^2\end{aligned}$$​​

Expressions

An expression is a string of terms separated by a $$+$$ or $$-$$.

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Examples

Simplifying expressions

Expressions can be simplified by adding or subtracting various terms. Terms with different powers cannot be added or subtracted.

Note: In order to be combined, two terms should be made of the same letter or same combination of letters.

Examples

​$$\begin{aligned}4x+5x&=9x\\ \ \\7y-3y&=4y\\ \ \\p+p+2p&=4p\\ \ \\3x^2+2x&=3x^2+2x\\ \ \\3x^2+2x-x^2-7x&=2x^2-5\\ \ \\4xy+3x-2xy+5y&=2xy+3x+5y\end{aligned}$$​​

Note: if a term has no number in front of it, it is assumed to be $$1$$. In the example above, $$p$$ means $$1p$$. The number $$1$$ is usually omitted.​

It is possible to simplify expressions to help us solve geometric problems.

Example 1

Write an expression for the perimeter of the following shape.

	$$m$$​​
$$n$$​​

Perimeter $$= \underline{2m+2n}$$

Example 2

Write an expression for the perimeter of the following shape. Simplify your answer as much as possible.

$$x+y$$​​		$$x+y$$​​
$$3x-y$$​​

Perimeter $$=x+y+x+y+3x-y$$

Perimeter $$=\underline{5x+2y}$$

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