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Collecting Like Terms
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Foundational Skills
Identify like terms in a pair
\[ \text{Are } 3x \text{ and } 5x \text{ like?} \]
Recognise whether two terms are like terms or not.
Identify like terms from a list
\[ \text{Which are like } 2x \text{?} \]
Select all terms from a list that are like a given term.
Sort terms into groups
\[ 3x, 2y, 5x, 4 \rightarrow \text{groups} \]
Group a set of terms by identifying which are like terms.
Explain why terms are unlike
\[ \text{Why } 3x \neq 3x^2 \text{?} \]
Articulate the reason why two terms cannot be combined.
Single Variable – Addition
Add two like terms
\[ 3x + 5x = 8x \]
Add two terms with the same variable.
Add including coefficient of 1
\[ x + 4x = 5x \]
Add two terms where one has an implicit coefficient of 1.
Single Variable – Subtraction
Subtract like terms (positive)
\[ 8x – 3x = 5x \]
Subtract like terms where the result is positive.
Subtract like terms (negative)
\[ 3x – 7x = -4x \]
Subtract like terms where the result is negative.
Subtract a term with coefficient 1
\[ 5x – x = 4x \]
Subtract a term where the second has implicit coefficient 1.
Subtract from a single variable
\[ x – 3x = -2x \]
Subtract a term from a variable with implicit coefficient 1.
Single Variable – Mixed Operations
Three terms (add and subtract)
\[ 5x + 2x – 3x = 4x \]
Combine three like terms using both + and −.
Four or more terms
\[ 3x + 5x – 2x + x \]
Combine four or more like terms.
Result is 1 or -1
\[ 5x – 3x – x = x \]
Combine like terms where the result has coefficient 1 or -1.
Multiple Variables – All Positive
Two variables (all positive)
\[ 3x + 2y + 5x + 4y \]
Collect like terms with two different variables.
Variables and constants
\[ 2x + 5 + 3x + 2 \]
Collect like terms including a constant term.
Three types (x, y, constant)
\[ 2x + 3y + x + 4 + 2y + 1 \]
Collect like terms with x-terms, y-terms and constants.
Multiple Variables – Mixed Signs
Two variables (mixed signs)
\[ 5x – 2y + 3x + 4y \]
Collect like terms with two variables including subtraction.
Variables and constants (mixed)
\[ 4x – 3 + 2x + 7 \]
Collect like terms with variables and constants including subtraction.
Three types (mixed signs)
\[ 3x – 2y + 4 – x + 5y – 1 \]
Collect like terms with x, y and constants with mixed signs.
Negative coefficient result
\[ 2x + 3y – 5x + y \]
Simplify where one variable ends up with a negative coefficient.
With Powers
Identify like terms with powers
\[ \text{Are } 3x^2 \text{ and } 5x^2 \text{ like?} \]
Recognise that terms need the same power to be like.
Collect terms with same power
\[ 3x^2 + 5x^2 = 8x^2 \]
Collect terms with the same variable and same power.
Mix x² and x terms
\[ 4x^2 + 3x + 2x^2 + 5x \]
Simplify an expression with both squared and linear terms.
x², x and constant
\[ 2x^2 + 3x + 1 + x^2 – x + 4 \]
Simplify with squared terms, linear terms and constants.
Unlike terms remain separate
\[ 5x + 3x^2 + 2x – x^2 \]
Recognise which terms combine and which don’t.
Special Cases
Terms that cancel to zero
\[ 4x + 3y – 4x + 2y \]
Simplify where one variable type cancels completely.
All terms cancel to zero
\[ 3x – 5x + 2x = 0 \]
Simplify where all terms cancel to give zero.
Result is just a constant
\[ 2x + 5 – 2x + 3 = 8 \]
Simplify where variable terms cancel leaving only a constant.
Result is negative
\[ 2x – 7x + 3 \]
Simplify with a negative coefficient in the answer.
Expression with ab-type terms
\[ 3ab + 2ab + ab \]
Collect like terms with two variables multiplied together.
Expression with ab and ba
\[ 2ab + 3ba + ab \]
Recognise that ab and ba are like terms.
Ordering the answer
\[ 3 + 2x + 5 – x \]
Write the simplified answer in conventional order.
Negative leading term
\[ -2x + 5x – 4x \]
Simplify an expression that starts with a negative term.
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