Simplify algebraic expressions

Ontario Grade 9 Mathematics

Algebra

Ontario MTH1W – Strand C: Algebra, Expectation C1.4

Imagine you are helping to organise a school fundraiser. You buy

x

boxes of granola bars and your friend buys

3

more boxes of the same kind, plus

2

bags of popcorn. Together you have

x + 3

boxes of granola bars and

2

bags of popcorn. You cannot combine the boxes and the bags into one neat number — they are different things. But you can combine all the granola-bar boxes into one group:

x + 3

boxes. That idea — grouping things of the same type — is exactly what simplifying algebraic expressions means. In this lesson you will learn the rules for deciding which parts of an expression can be combined, and you will practise simplifying expressions step by step.

What you will learn

Identify terms, coefficients, and like terms in an algebraic expression.
Collect like terms to simplify expressions involving addition and subtraction.
Apply the distributive property to expand and then simplify expressions.
Recognize when an expression is fully simplified and cannot be reduced further.

Prerequisites: Terms, Coefficients, and Variables

Before simplifying, you need to be able to read an algebraic expression correctly. An expression is a mathematical phrase made of numbers, variables, and operations. It does not have an equals sign.

A term is a single number, a single variable, or a number multiplied by one or more variables. Terms are separated from each other by addition or subtraction signs. For example, in the expression

3x + 5y - 7 + 2x^2

, the four terms are

3x

,

5y

,

-7

, and

2x^2

.

The coefficient is the number that is multiplied by the variable in a term. In the term

3x

, the coefficient is

3

. In the term

-7y

, the coefficient is

-7

. A term with no visible number in front, such as

x

, has a coefficient of

1

because

x = 1 \cdot x

. A constant term, such as

-7

, is a term with no variable at all.

The variable is the letter that represents an unknown or changing value. The exponent on a variable tells you the power to which it is raised. For example, in

2x^2

the variable is

x

and the exponent is

2

.

An expression has no equals sign; it is a phrase, not a sentence.
Terms are separated by $+$ or $-$ signs.
The coefficient is the number factor of a term.
A lone variable like $x$ has a coefficient of $1$ .

Like Terms: The Key to Combining

Like terms are terms that have exactly the same variable part — the same letter(s) raised to the same exponent(s). Only the coefficients may differ. Think of it this way:

3x

and

5x

are both 'groups of

x

', so they are like terms. But

3x

and

5x^2

are not like terms, because

x

and

x^2

represent different things, just as apples and apple crates are different things.

Constants (plain numbers with no variable) are always like terms with each other. So

-7

and

+4

are like terms because they are both plain numbers.

You can only add or subtract like terms. You cannot simplify

3x + 5y

any further because

x

and

y

are different variables — they represent different unknown quantities.

When you collect like terms, you add or subtract their coefficients and keep the variable part unchanged. For example,

3x + 5x = (3 + 5)x = 8x

. You are simply counting how many

x

's you have in total.

ax + bx = (a + b)x

Like terms share the exact same variable(s) and exponent(s).
Constants are like terms with other constants.
$3x$ and $3x^2$ are NOT like terms — different exponents make them different.
To collect like terms, add or subtract the coefficients only.
The variable part stays the same when you collect like terms.

Collecting Like Terms: Step-by-Step Method

The strategy for simplifying by collecting like terms has three steps: (1) Identify and label every term. (2) Group the like terms together. (3) Add or subtract the coefficients within each group and write the result.

Consider the expression

4a - 2b + 7a + 5b - 3

. First, identify all terms:

4a

,

-2b

,

7a

,

5b

, and

-3

. Second, group the like terms: the

a

-terms are

4a

and

7a

; the

b

-terms are

-2b

and

5b

; the constant is

-3

. Third, combine each group:

4a + 7a = 11a

;

-2b + 5b = 3b

; the constant stays as

-3

. The simplified expression is

11a + 3b - 3

.

Notice that you cannot combine

11a

,

3b

, and

-3

any further because they are all unlike terms. The expression is now fully simplified.

Sign management is important. When a subtraction sign appears in front of a term, that negative sign belongs to the term. Treat

-2b

as a term with coefficient

-2

, not as the number

2

being subtracted separately. This habit prevents the most common errors.

4a - 2b + 7a + 5b - 3 = 11a + 3b - 3

Step 1: Identify every term and its sign.
Step 2: Group like terms (rearranging is allowed, because addition is commutative).
Step 3: Add or subtract coefficients; keep the variable part unchanged.
A negative sign in front of a term belongs to that term's coefficient.
Stop when no two remaining terms are like terms.

The Distributive Property and Simplification

Sometimes an expression contains brackets. Before you can collect like terms, you must remove the brackets using the distributive property. The distributive property says: multiply the term outside the brackets by every term inside the brackets.

In plain language:

a(b + c)

means

a

copies of

(b + c)

, which is the same as

a

copies of

b

plus

a

copies of

c

. So

a(b + c) = ab + ac

.

When the number outside the bracket is negative, every sign inside the bracket flips. For example,

-3(2x - 5) = (-3)(2x) + (-3)(-5) = -6x + 15

. Expanding first and then collecting like terms is the reliable two-stage method.

Consider

2(3x + 4) - 5x + 1

. Expand the bracket:

6x + 8 - 5x + 1

. Now collect like terms:

x

-terms give

6x - 5x = x

; constants give

8 + 1 = 9

. The simplified result is

x + 9

.

a(b + c) = ab + ac

Expand brackets before collecting like terms.
Multiply the factor outside the bracket by every term inside.
A negative factor outside the bracket flips every sign inside.
After expanding, use the collecting-like-terms method to finish simplifying.

Recognising a Fully Simplified Expression

An expression is fully simplified when no two terms in it are like terms, and all brackets have been expanded. There is nothing left to combine.

For example,

5x^2 - 3x + 7

is fully simplified. The three terms

5x^2

,

-3x

, and

7

are all unlike: different variable parts and different exponents. You cannot add any of them together.

A common question is: 'Could I simplify

5x^2 - 3x

?' The answer is no. Even though both terms contain

x

, the exponents are different (

x^2

versus

x^1

), so they are unlike terms. You leave them as they are.

Knowing when to stop is just as important as knowing how to combine. Do one final check: scan every pair of terms and confirm none share the same variable part and exponent. If none do, you are done.

A fully simplified expression has no remaining like terms and no unexpanded brackets.
Different exponents on the same variable mean the terms are unlike — do not combine them.
Always do a final scan to confirm no two terms are still like terms.

Identifying Like and Unlike Terms at a Glance

Term	Variable Part	Exponent on Variable	Like Terms With…
$4x$	$x$	$1$	$-7x$ , $x$ , $12x$
$3x^2$	$x$	$2$	$-x^2$ , $5x^2$
$-2y$	$y$	$1$	$8y$ , $-y$
$6$	none (constant)	—	$-3$ , $10$ , $100$
$5xy$	$xy$	$1$ on each	$-xy$ , $2xy$

Worked example

Simplify an Expression with Brackets and Multiple Like Terms

Simplify the expression

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

.

Expand the first bracket
Multiply $3$ by every term inside $(2x - 4)$ . Multiplying $3$ by $2x$ gives $6x$ . Multiplying $3$ by $-4$ gives $-12$ .
$3(2x - 4) = 6x - 12$
Expand the second bracket
Multiply $-2$ by every term inside $(x + 3)$ . Multiplying $-2$ by $x$ gives $-2x$ . Multiplying $-2$ by $3$ gives $-6$ . The negative factor flips both signs.
$-2(x + 3) = -2x - 6$
Rewrite the expression with all brackets removed
Replace each bracketed group with its expanded form. Keep the $5x$ term exactly as it was — it had no bracket.
$6x - 12 + 5x - 2x - 6$
Group the like terms
Identify the $x$ -terms: $6x$ , $5x$ , and $-2x$ . Identify the constant terms: $-12$ and $-6$ . Rearranging the order of terms is allowed because addition can be done in any order.
$(6x + 5x - 2x) + (-12 - 6)$
Combine the x-terms
Add the coefficients $6 + 5 - 2 = 9$ . The variable part stays as $x$ .
$6x + 5x - 2x = 9x$
Combine the constant terms
Add the constants $-12 + (-6) = -18$ .
$-12 - 6 = -18$
Write the final simplified expression
Join the two results. There are no remaining like terms, so this is fully simplified.
$9x - 18$

Answer:

9x - 18

Check: Substitute

x = 2

into the original expression and into the answer to verify they are equal. Original:

3(2(2) - 4) + 5(2) - 2(2 + 3) = 3(0) + 10 - 2(5) = 0 + 10 - 10 = 0

. Answer:

9(2) - 18 = 18 - 18 = 0

. Both equal

0

, so the simplification is correct.

Common mistakes and how to avoid them

Combining unlike terms, such as writing

3x + 5x^2 = 8x^2

or

8x^3

.

Correction: Check the exponent on each term.

3x

has exponent

1

and

5x^2

has exponent

2

. They are unlike terms and cannot be combined. The expression stays as

3x + 5x^2

.

Forgetting to distribute to every term inside the bracket, such as writing

3(2x - 4) = 6x - 4

instead of

6x - 12

.

Correction: The factor outside the bracket must multiply every single term inside. Here,

3 \times 2x = 6x

AND

3 \times (-4) = -12

, giving

6x - 12

.

Dropping the negative sign when expanding a bracket, such as writing

-2(x + 3) = -2x + 3

instead of

-2x - 6

.

Correction: A negative factor outside the bracket flips the sign of every term inside.

-2 \times 3 = -6

, not

+3

. Always multiply the signs as well as the numbers.

Treating the subtraction sign as belonging to the operator rather than to the term, and then losing it during rearrangement.

Correction: Attach each sign to the term that follows it before rearranging. In

5a - 3b + 2a

, the

-3b

is a single term with coefficient

-3

. Keep that sign with

3b

at all times.

Thinking a simplified expression must be a single term, so attempting to combine

4x + 7

into one piece.

Correction:

4x

and

7

are unlike terms. A simplified expression can still have multiple terms. Stop simplifying when no remaining terms are alike.

Lesson summary

An algebraic expression is made of terms separated by $+$ or $-$ signs. Each term has a coefficient and a variable part.
Like terms share exactly the same variable(s) raised to the same exponent(s). Only like terms can be added or subtracted.
To collect like terms, add or subtract their coefficients and write the result with the same variable part unchanged.
When an expression contains brackets, use the distributive property to expand first, then collect like terms.
A negative factor outside a bracket flips the sign of every term inside when you expand.
An expression is fully simplified when all brackets are expanded and no two remaining terms are like terms.

Check your understanding

Question 1

Which pair of terms are like terms?

$4x$ and $4y$
$3x^2$ and $3x$
$-5x$ and $9x$
$2xy$ and $2x$

Show answer and explanation

-5x

and

9x

-5x

and

9x

are like terms because they both have the variable

x

raised to the power of

1

. The other pairs have different variables or different exponents, making them unlike terms.

Question 2

Simplify

6m - 4 + 2m + 9

.

$8m + 5$
$8m - 13$
$4m + 5$
$8m + 13$

Show answer and explanation

8m + 5

Collect the

m

-terms:

6m + 2m = 8m

. Collect the constants:

-4 + 9 = 5

. The simplified expression is

8m + 5

.

Question 3

Expand and simplify

4(x + 2) - 3x

.

$7x + 8$
$x - 8$
$x + 8$
$7x - 8$

Show answer and explanation

x + 8

Expand the bracket:

4(x + 2) = 4x + 8

. Rewrite the expression:

4x + 8 - 3x

. Collect like terms:

4x - 3x = x

. The constant

8

remains. The simplified expression is

x + 8

.

Question 4

Simplify

5a - 2(3a - 4) + 1

.

$-a + 9$
$-a - 7$
$11a - 7$
$-a + 4$

Show answer and explanation

-a + 9

Expand the bracket:

-2(3a - 4) = -6a + 8

. Rewrite:

5a - 6a + 8 + 1

. Collect

a

-terms:

5a - 6a = -a

. Collect constants:

8 + 1 = 9

. The simplified expression is

-a + 9

.

Key terms

Expression: A mathematical phrase containing numbers, variables, and operations, but no equals sign.
Term: A single number, variable, or product of numbers and variables. Terms are separated by addition or subtraction signs.
Coefficient: The number multiplied by the variable(s) in a term. For example, in $-7x$ , the coefficient is $-7$ .
Constant: A term that is a plain number with no variable, such as $5$ or $-3$ .
Like terms: Terms that have exactly the same variable(s) raised to exactly the same exponent(s). Only like terms can be combined by addition or subtraction.
Distributive property: The rule that $a(b + c) = ab + ac$ : multiply the factor outside the brackets by every term inside the brackets.
Simplify: To rewrite an expression in its most compact form by expanding all brackets and collecting all like terms, with no like terms remaining.
Exponent: The small raised number that tells how many times a base is multiplied by itself. In $x^2$ , the exponent is $2$ .

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation C1.4. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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C1.4 · Simplify algebraic expressions

Ontario Grade 9 Mathematics

Ontario MTH1W – Strand C: Algebra, Expectation C1.4

What you will learn

Prerequisites: Terms, Coefficients, and Variables

Like Terms: The Key to Combining

Collecting Like Terms: Step-by-Step Method

The Distributive Property and Simplification

Recognising a Fully Simplified Expression

Identifying Like and Unlike Terms at a Glance

Simplify an Expression with Brackets and Multiple Like Terms

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MTH1W

About this lesson and its review