Collecting Like Terms

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Collecting Like Terms

A term is an individual part of an expression and typically appears in one of three forms:

A number by itself
A letter by itself
A combination of letters and numbers

Like terms have the same combination of letters. To add or subtract terms with the same letter, we add or subtract the numbers like usual and just put the letter back on the end.

Skill 1: Linear Expressions with a Single Variable

Linear expressions are those in which all variables (i.e. the letters) are to the power of $1$ so there are no squared or cubed terms.

Example: simplify the following,

$2a + 3 + a + 5$

We have to group the terms that are similar; all the terms that are just numbers need grouping together and simplifying, as do the similar algebraic terms.

This gives,

$2a+1a = 3a$

$3 + 5 = 8$

This gives the final answer to be,

$2a + 3 + a + 5 = 3a + 8$

Note: Normally the $1$ in front of the $a$ is left out but we have included it here to make it clear the steps being taken.

Skill 2: Like Terms With Different Letters

Sometimes terms have more than one variable (letter) multiplied or divided together:

$textcolor{red}{xy} + textcolor{blue}{y} + textcolor{red}{2xy}$

in this instance, $textcolor{red}{xy}$ and $textcolor{red}{2xy}$ , are like terms that can be added together to simplify the expression, hence we find,

$textcolor{red}{3xy} + textcolor{blue}{y}$

Skill 3: Identifying Like Terms with Powers

Terms of certain powers have to be grouped with terms of the same power. Such terms can also include multiple letters and powers.

$2textcolor{red}{x^2y} + textcolor{blue}{xy^2} + 3textcolor{red}{x^2y} + 5textcolor{blue}{xy^2} = 5textcolor{red}{x^2y} + 6textcolor{blue}{xy^2}$

$2textcolor{red}{x^2y}$ and $3textcolor{red}{x^2y}$ are like terms
$textcolor{blue}{xy^2}$ and $5textcolor{blue}{xy^2}$ are like terms

Example 1: Two Variables

Simplify the expression $3y+2x+4x-y$ .

[1 mark]

$textcolor{blue}{3y}$ and $-textcolor{blue}{y}$ are like terms

$textcolor{purple}{2x}$ and $textcolor{purple}{4x}$ are like terms

Collecting Like Terms Adding and Subtracting Two Variables

Example 2: Including Powers

Simplify the expression $4x^2 +2x +3x^2$ .

[1 mark]

$textcolor{limegreen}{4x^2}$ and $textcolor{limegreen}{3x^2}$ are like terms

$textcolor{purple}{2x}$ has no like terms

Example 3: Multiple Letters

Simplify the expression $5mn +3xy+mn+x$ .

[1 mark]

$textcolor{maroon}{5mn}$ and $textcolor{maroon}{mn}$ are like terms

$textcolor{blue}{3xy}$ and $textcolor{limegreen}{x}$ have no other like terms

Example 4: Multiple Letters and Powers

Collecting Like Terms Multiple Letters and Powers

Simplify the expression $2m^2n+3mn+2m^2n$ .

[1 mark]

$textcolor{maroon}{2m^2n}$ and $textcolor{maroon}{2m^2n}$ are like terms

$textcolor{blue}{3mn}$ has no like terms

Collecting Like Terms Example Questions

Question 1: Simplify the following expression by collecting like terms:

[1 mark]

$5x+5-2x+3-4-x$

It is often easier to first group the similar terms together, before simplifying,

$5x+5-2x+3-4-x =(5x-2x-x)+(5+3-4) =2x+4$

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Question 2: Simplify the following expression by collecting like terms:

[1 mark]

$ab+bc+2ab-bc+a$

It is often easier to first group the similar terms together, before simplifying,

$ab+bc+2ab-bc+a = a+(ab+2ab)+(bc-bc) = a+3ab$

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Question 3: Simplify the following expression by collecting like terms:

[1 mark]

$11x+7y-2x-13y$

It is often easier to first group the similar terms together, before simplifying:

$11x+7y-2x-13y = (11x-2x)+(7y-13y) =9x-6y$

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Question 4: Simplify the following expression by collecting like terms:

[1 mark]

$2m+6n-3+8n +5m$

It is often easier to first group the similar terms together, before simplifying:

$2m+6n-3+8n+5n = (2m+5m)+(6n+8n)-3 = 7m+14n-3$

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Question 5: Simplify the following expression by collecting like terms:

[1 mark]

$2a^2 + 5b-2a -3b+5a^2$

It is often easier to first group the similar terms together, before simplifying:

$2a^2 + 5b-2a -3b+5a^2 = (2a^2+5a^2)+(5b-3b)-2a =7a^2+2b-2a$

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Specification Points Covered

Algebra – 1. use and interpret algebraic notation, including:

$ab$ in place of $a times b$
$3y$ in place of $y+y+y$ and $3 times y$
$a^2$ in place of $a times a$ , $a^3$ in place of $a times a times a$ , $a^2 b$ in place of $a times a times b$
$dfrac{a}{b}$ in place of $a div b$
coefficients written as fractions rather than as decimals
brackets

Algebra – 4. simplify and manipulate algebraic expressions (including those involving surds and algebraic fractions) by:

collecting like terms
multiplying a single term over a bracket
taking out common factors
expanding products of two or more binomials
factorising quadratic expressions of the form $x^2 + bx + c$ , including the
difference of two squares; factorising quadratic expressions of the form $ax^2 + bx + c$
simplifying expressions involving sums, products and powers, including the laws of indices