Compare expressions and identify equivalent forms

Ontario Grade 9 Mathematics

Algebra

Ontario MTH1W — Strand C: Algebra, Expectation C1.3

You already know how to write and evaluate simple algebraic expressions from Grade 8. In this lesson you will go one step further: you will compare expressions to decide whether two that look different are actually saying the exact same thing. This skill matters every time you rearrange a formula, check someone else's algebra, or choose the most useful way to write an answer. By the end of this lesson you will be able to look at two expressions, test them, simplify them, and state with confidence whether they are equivalent.

What you will learn

Explain what it means for two algebraic expressions to be equivalent.
Simplify expressions by collecting like terms and applying the distributive property.
Compare two or more expressions by substituting values and by simplifying, then decide whether they are equivalent.
Identify different equivalent forms of the same expression and explain why they are equal.

What Does 'Equivalent' Mean?

An expression is a mathematical phrase that contains numbers, variables, or both, combined with operations. Examples:

3x + 2

,

5(x - 1)

,

2x + x + 2

. An expression does not have an equals sign — it is not an equation.

Two expressions are called equivalent expressions when they produce the same output for every value of the variable. Think of them as two different recipes that always cook the same meal, no matter which ingredient amounts you choose.

Concrete start: Imagine you and a friend each count the tiles around a square pool. You write the count as

4(n + 1)

, where

n

is the number of tiles along one side of the pool. Your friend writes

4n + 4

. Are these the same count? Try

n = 3

: your expression gives

4(3 + 1) = 4 \times 4 = 16

. Your friend's gives

4(3) + 4 = 12 + 4 = 16

. Try

n = 10

: yours gives

4(11) = 44

; your friend's gives

40 + 4 = 44

. Same result every time — so these two expressions are equivalent.

4(n + 1) = 4n + 4

An expression contains numbers and variables combined with operations, but no equals sign.
Two expressions are equivalent when they give the same value for every substitution of the variable.
Equivalent expressions can look very different but always produce matching outputs.

Two Methods for Comparing Expressions

There are two main methods to decide whether expressions are equivalent. The first is substitution testing: replace the variable with several different numbers and check whether both expressions give the same result each time. This is a quick check, but it cannot prove equivalence on its own — it is possible (though unlikely with multiple test values) that two non-equivalent expressions match at one particular number by coincidence. Use at least two or three different values, including a negative number, to make your substitution test reliable.

The second method is algebraic simplification: rewrite each expression in its simplest form using rules you already know — collecting like terms and the distributive property — and then compare those simplified forms. If both expressions simplify to the exact same form, they are equivalent. This method is more certain than substitution alone.

Like terms are terms that have exactly the same variable part. For example,

5x

and

-2x

are like terms (both have

x

), but

5x

and

5x^2

are not like terms (different powers). You can only add or subtract like terms. The distributive property says

a(b + c) = ab + ac

: multiply the term outside the brackets by every term inside.

a(b + c) = ab + ac

Method 1 — Substitution: replace the variable with numbers and compare outputs. Use at least two or three test values.
Method 2 — Simplification: collect like terms and apply the distributive property; compare the resulting simplified forms.
Like terms share exactly the same variable part and can be combined.
The distributive property: $a(b + c) = ab + ac$ .

Simplifying to Find Equivalent Forms

Collecting like terms means grouping terms with the same variable, then adding or subtracting their coefficients. The coefficient is the number in front of the variable. For example, in

7x

, the coefficient is

7

.

Consider

3x + 5 + 2x - 1

. Group the

x

-terms:

3x + 2x = 5x

. Group the constant terms (plain numbers):

5 - 1 = 4

. The simplified form is

5x + 4

. Any other expression that also simplifies to

5x + 4

is equivalent to the original.

Now apply the distributive property first, then collect like terms. Consider

2(x + 3) + 3x - 1

. Distribute:

2 \cdot x + 2 \cdot 3 + 3x - 1 = 2x + 6 + 3x - 1

. Collect like terms:

2x + 3x = 5x

and

6 - 1 = 5

, giving

5x + 5

. Is

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

equivalent? Distribute:

3x + 6 - x + x - 1

. Collect:

3x - x + x = 3x

and

6 - 1 = 5

, giving

3x + 5

. Since

5x + 5 \neq 3x + 5

(the coefficients of

x

differ), these two expressions are NOT equivalent.

3x + 5 + 2x - 1 = 5x + 4

Collect like terms by adding or subtracting their coefficients.
Apply the distributive property before collecting like terms when brackets are present.
Two expressions are equivalent only if their fully simplified forms are identical.
Always check both the variable part and the constant when comparing simplified forms.

Comparing Multiple Forms of the Same Expression

One expression can have many equivalent forms. The form you choose often depends on what you want to do with it. For example, the expression

2x + 6

and the expression

2(x + 3)

are equivalent — you can verify this by distributing:

2(x + 3) = 2x + 6

. The form

2(x + 3)

shows clearly that the expression is 2 times a group, which can be useful for spotting patterns. The form

2x + 6

is expanded and may be easier for substitution. Both forms are correct; neither is wrong.

When a question asks you to identify equivalent forms from a list, your strategy is: simplify every expression in the list, then group the ones with matching simplified forms. All expressions in one group are equivalent to each other.

Here is an important boundary: two expressions are equivalent only if they match for ALL values of the variable, not just one or two. If even one substitution gives a different result, the expressions are not equivalent. For example,

x^2

and

2x

both equal

4

when

x = 2

, but when

x = 3

,

x^2 = 9

and

2x = 6

. They are not equivalent, even though one test value matched.

2(x + 3) = 2x + 6

A single expression can be written in many equivalent forms.
To compare a list of expressions, simplify each one and match those with identical simplified forms.
Two expressions must agree for every value of the variable to be considered equivalent.
A single matching substitution value does not prove equivalence — always verify with simplification or multiple test values.

Substitution Test: Comparing $4(x+2)-x+1$, $3x+9$, and $3(x+3)$

Value of $x$	Expression A: $4(x+2)-x+1$	Expression B: $3x+9$	Expression C: $3(x+3)$
$x = 0$	$4(2)-0+1=9$	$0+9=9$	$3(3)=9$
$x = 2$	$4(4)-2+1=15$	$6+9=15$	$3(5)=15$
$x = -1$	$4(1)-(-1)+1=6$	$-3+9=6$	$3(2)=6$
$x = 5$	$4(7)-5+1=24$	$15+9=24$	$3(8)=24$

Worked example

Are These Three Expressions Equivalent?

Three students each wrote an expression for the perimeter of a shape. Decide which expressions are equivalent. Expression A:

4(x + 2) - x + 1

. Expression B:

3x + 9

. Expression C:

3(x + 3)

.

Step 1 — Simplify Expression A
Apply the distributive property to expand $4(x + 2)$ , then collect all like terms across the whole expression.
$4(x + 2) - x + 1 = 4x + 8 - x + 1$
Step 1 continued — Collect like terms in A
Group the $x$ -terms together and the constant terms together, then add each group.
$4x - x + 8 + 1 = 3x + 9$
Step 2 — Simplify Expression B
Expression B is already fully simplified — there are no like terms to collect and no brackets to expand.
$3x + 9$
Step 3 — Simplify Expression C
Apply the distributive property to expand $3(x + 3)$ .
$3(x + 3) = 3x + 9$
Step 4 — Compare all three simplified forms
All three simplified forms are identical, so all three expressions are equivalent to each other.
$3x + 9 = 3x + 9 = 3x + 9$
Step 5 — Verify with a substitution
Substitute $x = 2$ into each original expression to confirm. All three should give the same number.
$4(2+2)-2+1 = 15, \ 3(2)+9 = 15, \ 3(2+3) = 15$

Answer: All three expressions are equivalent. Each simplifies to

3x + 9

.

Check: Substituting

x = 0

into each: Expression A gives

4(2) - 0 + 1 = 9

; Expression B gives

0 + 9 = 9

; Expression C gives

3(3) = 9

. All match, which supports the conclusion.

Common mistakes and how to avoid them

Concluding two expressions are equivalent after testing only one value of the variable.

Correction: One matching result is not enough. Always test at least two or three values, including a negative number, or use algebraic simplification to be certain.

Forgetting to distribute to every term inside the brackets, e.g., writing

3(x + 4) = 3x + 4

instead of

3x + 12

.

Correction: Multiply the term outside the brackets by every term inside:

3(x + 4) = 3 \cdot x + 3 \cdot 4 = 3x + 12

.

Combining terms that are not like terms, such as adding

5x

and

3

to get

8x

.

Correction:

5x

and

3

are unlike terms — one has a variable, one does not. They cannot be combined: the simplified form stays as

5x + 3

.

Losing a negative sign when collecting like terms, e.g., treating

4x - x

as

4x + x = 5x

instead of

3x

.

Correction: Keep the sign attached to the term that follows it.

4x - x

means

4x + (-1x) = 3x

.

Thinking the form of an expression matters for equivalence — for example, believing

2(x+3)

and

2x+6

are different expressions because they look different.

Correction: Equivalent expressions can look different. What matters is whether they always produce the same output, not whether they use the same notation or form.

Lesson summary

Two expressions are equivalent when they produce the same value for every possible substitution of the variable.
To compare expressions, either substitute multiple values (including at least one negative) or simplify each expression fully and compare the results.
Simplification uses two main tools: collecting like terms (combining terms with the same variable part) and the distributive property ( $a(b+c) = ab + ac$ ).
A single matching substitution value does not confirm equivalence — use simplification or multiple test values to be sure.
One expression can have many equivalent forms; choosing among them depends on what is most useful in context.
If two fully simplified expressions are not identical, the original expressions are not equivalent — even if they matched at one test value.

Check your understanding

Question 1

Which expression is equivalent to

5x - 2x + 8

?

$3x + 8$
$7x + 8$
$3x - 8$
$5x + 6$

Show answer and explanation

3x + 8

Collect the like terms:

5x - 2x = 3x

. The constant

8

stays as is. The simplified form is

3x + 8

.

Question 2

A student says

2(x + 5)

and

2x + 5

are equivalent. Which substitution best shows this is wrong?

$x = 0$ , giving $10 \neq 5$
$x = 1$ , giving $1 \neq 1$
$x = 5$ , giving $20 = 20$
$x = -5$ , giving $0 = 0$

Show answer and explanation

x = 0

, giving

10 \neq 5

At

x = 0

:

2(0+5) = 10

but

2(0)+5 = 5

. Since

10 \neq 5

, the expressions are not equivalent. This single counterexample is enough to disprove the claim.

Question 3

Which two expressions from this list are equivalent to each other? (i)

3(x + 2) + x

(ii)

4x + 6

(iii)

4x + 2

(i) and (ii)
(i) and (iii)
(ii) and (iii)
All three are equivalent

Show answer and explanation

(i) and (ii)

Simplify (i):

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

. Expression (ii) is already

4x+6

. Expression (iii) is

4x+2

, which differs in the constant. So (i) and (ii) are equivalent.

Question 4

Which statement best describes why

x^2 + x

and

2x

are NOT equivalent?

They give different values for most substitutions, such as $x = 3$ where $x^2+x=12$ but $2x=6$ .
They look different, so they cannot be equivalent.
They give the same value only when $x = 0$ , which proves they are not equivalent for any value.
You cannot compare expressions that have $x^2$ in them.

Show answer and explanation

They give different values for most substitutions, such as

x = 3

where

x^2+x=12

but

2x=6

.

At

x = 3

:

x^2 + x = 9 + 3 = 12

and

2x = 6

. Since

12 \neq 6

, the expressions are not equivalent. The appearance of expressions is not the reason — we always check values or simplify to decide.

Key terms

Expression: A mathematical phrase made up of numbers, variables, and operations (such as addition or multiplication). It does not contain an equals sign.
Equivalent expressions: Two or more expressions that produce exactly the same value for every possible value of the variable.
Variable: A letter (such as $x$ or $n$ ) that stands for a number that can change.
Coefficient: The number multiplied by the variable in a term. In $7x$ , the coefficient is $7$ .
Like terms: Terms that have exactly the same variable part. For example, $4x$ and $-x$ are like terms; $4x$ and $4x^2$ are not.
Collecting like terms: Combining like terms by adding or subtracting their coefficients to produce a simpler expression.
Distributive property: The rule that $a(b + c) = ab + ac$ : multiply the factor outside the brackets by every term inside.
Substitution: Replacing a variable with a specific number in order to calculate the value of the expression.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation C1.3. 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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C1.3 · Compare expressions and identify equivalent forms

Ontario Grade 9 Mathematics

Ontario MTH1W — Strand C: Algebra, Expectation C1.3

What you will learn

What Does 'Equivalent' Mean?

Two Methods for Comparing Expressions

Simplifying to Find Equivalent Forms

Comparing Multiple Forms of the Same Expression

Substitution Test: Comparing $4(x+2)-x+1$, $3x+9$, and $3(x+3)$

Are These Three Expressions Equivalent?

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