# 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 $-$.

##### Examples

$2x+2y$ |

$a-2b$ |

$x^2-4x+4$ |

$xy+x^2+3y+1$ |

## 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.*

*Perimeter *$= \underline{2m+2n}$

##### Example 2

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

*Perimeter *$=x+y+x+y+3x-y$

*Perimeter *$=\underline{5x+2y}$

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