Determine whether algebraic expressions are equivalent

Ontario Grade 11 Mathematics

Characteristics of Functions

MCR3U · Strand A · Expectation A3.4

Two expressions that look completely different can secretly produce the same output for every possible input. Recognising when this is true — and when it is not — is a core algebraic skill that runs through the whole MCR3U course. In this lesson you will learn a reliable two-pronged strategy: simplify both expressions fully and compare, then verify with substitution. By the end you will be able to decide with confidence whether any two algebraic expressions are equivalent.

What you will learn

Explain what it means for two algebraic expressions to be equivalent.
Use simplification and substitution to test whether two expressions are equivalent.
Identify when a single substitution test is not enough to confirm equivalence.
Determine equivalence for polynomial, rational, and other algebraic expressions encountered in MCR3U.

Prerequisite Bridge: What You Already Know

Before testing equivalence you need two Grade 10 skills: expanding brackets using the distributive law, and collecting like terms. A quick reminder: the distributive law says

a(b + c) = ab + ac

. Like terms share the same variable part, so

3x^2

and

-7x^2

are like terms, but

3x^2

and

3x

are not.

You also need to remember how to factor simple expressions. For example,

x^2 - 9

factors as

(x-3)(x+3)

because it is a difference of squares. These tools are the building blocks for everything that follows.

If either of these feels rusty, pause here and review before continuing. The rest of the lesson builds directly on them.

Distributive law: $a(b+c) = ab + ac$ .
Like terms share the same variable and exponent.
Difference of squares: $a^2 - b^2 = (a-b)(a+b)$ .

What Does Equivalence Mean?

Two algebraic expressions are called equivalent if they produce exactly the same output for every value of the variable(s) where both expressions are defined. The key word is every. One matching value is not enough.

Think of it like two different-looking vending machines that always dispense the same snack no matter which button you press. The machines look different on the outside, but the result is always identical.

Equivalence is written with the equals sign between two simplified forms, for example

3(x + 2) = 3x + 6

. This equation is true for all real values of

x

, so the two sides are equivalent expressions. Contrast this with an equation like

x + 5 = 9

, which is only true for

x = 4

. That is a conditional equation, not an equivalence.

In MCR3U you will encounter polynomial expressions, rational expressions (fractions with variables), and expressions involving exponents. The definition of equivalence is the same in every case, but the simplification tools you use will vary.

Equivalent expressions give the same output for every allowed input value.
Equivalence is different from a conditional equation, which is only true for specific values.
The word 'defined' matters: a rational expression is undefined where its denominator equals zero.

Two Reliable Methods for Testing Equivalence

Method 1 — Full Simplification: Simplify each expression independently until it is in its simplest form. If both simplified forms are identical, the original expressions are equivalent. This method gives a definitive answer and shows your algebraic reasoning clearly. It is the primary method you should use.

Method 2 — Substitution Check: Substitute a convenient number for the variable into both expressions. If the outputs differ, the expressions are definitely not equivalent — you have found a counterexample, and one counterexample is enough to disprove equivalence. If the outputs match, this is promising but not conclusive on its own, because two non-equivalent expressions can occasionally agree at a single point.

The safest strategy combines both methods: simplify first, then confirm with substitution. If your algebra says 'equivalent' and a substitution also agrees, you can be confident. If your algebra says 'not equivalent', one substitution that gives different outputs seals the case.

Full simplification gives a definitive conclusion.
One substitution that gives different outputs disproves equivalence immediately.
One substitution that gives matching outputs does not prove equivalence on its own.
Use both methods together for the strongest argument.

Simplifying Rational Expressions to Test Equivalence

A rational expression is a fraction whose numerator and denominator are polynomials, for example

\frac{x^2 - 4}{x - 2}

. To simplify a rational expression, factor the numerator and denominator, then cancel any common factors — this is exactly the same process as simplifying a numerical fraction like

\frac{6}{4} = \frac{3}{2}

.

An important restriction: whenever you cancel a factor from the denominator you must state that the original expression is undefined at the value that makes that factor zero. For example,

\frac{x^2 - 4}{x - 2} = \frac{(x-2)(x+2)}{x-2} = x + 2

, but only for

x \neq 2

. The simplified form

x + 2

is defined at

x = 2

, but the original fraction is not, so they are not completely equivalent without that restriction.

When comparing two rational expressions for equivalence, simplify both, state any restrictions, and then compare the resulting polynomials. If the simplified polynomials match and the restrictions are the same, the expressions are equivalent.

\frac{a · c}{b · c} = \frac{a}{b}, b ≠ 0,\ c ≠ 0

Factor numerator and denominator, then cancel common factors.
State restrictions: values of the variable that make any denominator zero.
Two rational expressions are equivalent only when their simplified forms and restrictions match.

Putting It All Together: Strategy and Common Pitfalls

Here is the full strategy in plain steps. First, simplify each expression completely — expand brackets, collect like terms, and for rational expressions factor and cancel. Second, compare the simplified forms. Third, substitute one or two numbers into both original expressions to double-check. Fourth, state your conclusion with a reason.

A common pitfall is cancelling terms instead of factors. You can only cancel a factor that multiplies the entire numerator with a factor that multiplies the entire denominator. For instance,

\frac{x^2 + 4x}{x}

simplifies to

x + 4

because

x

is a factor of the whole numerator:

\frac{x(x+4)}{x} = x + 4

. But

\frac{x^2 + 4}{x}

cannot be simplified to

x + 4

because

x

does not divide evenly into

x^2 + 4

.

Another pitfall is testing equivalence at

x = 0

or

x = 1

only. These values are convenient but can hide differences. For example, the expressions

x

and

x^2

both equal

0

when

x = 0

and both equal

1

when

x = 1

, yet they are clearly not equivalent for other values. Always test at a less obvious value, such as

x = 3

or

x = -2

, as a second check.

Cancel factors, not terms.
Avoid testing only at $x = 0$ or $x = 1$ ; use a less obvious value as well.
State restrictions for any rational expression.
A clear written conclusion — 'equivalent' or 'not equivalent' with a reason — is part of a complete answer.

Substitution Test Results: Are x and x² Equivalent?

Value of x	Output of x	Output of x²	Same output?
0	0	0	Yes — misleading!
1	1	1	Yes — misleading!
3	3	9	No — not equivalent
-2	-2	4	No — not equivalent

Worked example

Example 1 — Polynomial Expressions (Mixed Difficulty)

Determine whether the expressions

P = 2(x^2 + 3x - 1) - (x^2 - x + 4)

and

Q = (x + 5)(x + 1) - 9

are equivalent. Show full simplification and verify with a substitution.

Simplify P — distribute the coefficients
Apply the distributive law to each bracket in $P$ . Multiply every term inside the first bracket by $2$ , and every term inside the second bracket by $-1$ (the minus sign out front).
$P = 2x^2 + 6x - 2 - x^2 + x - 4$
Simplify P — collect like terms
Group the $x^2$ terms, the $x$ terms, and the constant terms, then add them.
$P = (2x^2 - x^2) + (6x + x) + (-2 - 4) = x^2 + 7x - 6$
Simplify Q — expand the brackets using FOIL
Multiply $(x+5)(x+1)$ by multiplying each term in the first bracket by each term in the second bracket: $x \cdot x + x \cdot 1 + 5 \cdot x + 5 \cdot 1$ , then subtract $9$ .
$Q = x^2 + x + 5x + 5 - 9 = x^2 + 6x - 4$
Compare the simplified forms
Write the two simplified results side by side. $P$ simplifies to $x^2 + 7x - 6$ and $Q$ simplifies to $x^2 + 6x - 4$ . The coefficients of $x$ differ ( $7$ vs $6$ ) and the constants differ ( $-6$ vs $-4$ ), so the expressions are not equivalent.
$x^2 + 7x - 6 ≠ x^2 + 6x - 4$
Verify with a substitution
Substitute $x = 2$ into both simplified forms to confirm the difference. For $P$ : $4 + 14 - 6 = 12$ . For $Q$ : $4 + 12 - 4 = 12$ . These agree! This shows why one test value is not enough — try $x = 3$ instead. For $P$ : $9 + 21 - 6 = 24$ . For $Q$ : $9 + 18 - 4 = 23$ . The outputs differ, confirming non-equivalence.
$P(3) = 24 \ \ \ \ Q(3) = 23$

Answer: The expressions are not equivalent.

P

simplifies to

x^2 + 7x - 6

and

Q

simplifies to

x^2 + 6x - 4

, which are different polynomials. The substitution

x = 3

confirms this:

P = 24

while

Q = 23

.

Check: Re-expanding P:

2x^2 + 6x - 2 - x^2 + x - 4 = x^2 + 7x - 6

. ✓ Re-expanding Q:

x^2 + 6x + 5 - 9 = x^2 + 6x - 4

. ✓ These are clearly not the same polynomial.

Worked example

Example 2 — Rational Expression (Harder)

Determine whether

R = \frac{3x^2 + 6x}{x^2 - x - 6}

and

S = \frac{3x}{x - 3}

are equivalent. Simplify fully, state any restrictions, and verify with a substitution.

Factor the numerator of R
Look for a common factor in the numerator $3x^2 + 6x$ . Both terms share a factor of $3x$ , so factor it out.
$3x^2 + 6x = 3x(x + 2)$
Factor the denominator of R
Factor $x^2 - x - 6$ by finding two numbers that multiply to $-6$ and add to $-1$ . Those numbers are $-3$ and $2$ .
$x^2 - x - 6 = (x - 3)(x + 2)$
State restrictions for R before cancelling
The denominator $(x-3)(x+2)$ equals zero when $x = 3$ or $x = -2$ . Both values must be excluded from the domain of $R$ .
$x ≠ 3, x ≠ -2$
Cancel the common factor in R
The factor $(x + 2)$ appears in both the numerator and denominator. Cancel it, keeping in mind the restriction $x \neq -2$ still applies even after cancelling.
$R = \frac{3x(x+2)}{(x-3)(x+2)} = \frac{3x}{x-3}, x ≠ 3,\ x ≠ -2$
State restrictions for S
The expression $S = \frac{3x}{x-3}$ has denominator $x - 3$ , which equals zero when $x = 3$ . So the only restriction for $S$ is $x \neq 3$ .
$x ≠ 3$
Compare simplified forms and restrictions
After simplification, both $R$ and $S$ have the same polynomial expression $\frac{3x}{x-3}$ . However, $R$ has two restrictions ( $x \neq 3$ and $x \neq -2$ ) while $S$ has only one ( $x \neq 3$ ). Because the domains differ, $R$ and $S$ are not fully equivalent.
$R: \frac{3x}{x-3},\ x ≠ 3,\ x ≠ -2 \text{vs} S: \frac{3x}{x-3},\ x ≠ 3$
Verify with a substitution
Substitute $x = -2$ into both originals. For $R$ : the denominator is $(-2)^2 - (-2) - 6 = 4 + 2 - 6 = 0$ , so $R$ is undefined. For $S$ : $\frac{3(-2)}{-2-3} = \frac{-6}{-5} = \frac{6}{5}$ , which is defined. This confirms $R$ and $S$ behave differently at $x = -2$ .
$R(-2) = \text{undefined}, S(-2) = \frac{6}{5}$

Answer: R and S are not equivalent. Although they simplify to the same fraction

\frac{3x}{x-3}

, the original expression

R

is undefined at

x = -2

while

S

is defined there. Equivalent expressions must agree for every value in their domain, and here they do not.

Check: Numerator of R factored:

3x(x+2)

. ✓ Denominator of R factored:

(x-3)(x+2)

. ✓ After cancelling

(x+2)

:

\frac{3x}{x-3}

with restrictions

x \neq 3, x \neq -2

. ✓ At

x = -2

, R is undefined and S gives

\frac{6}{5}

. ✓ Conclusion confirmed.

Common mistakes and how to avoid them

Cancelling terms instead of factors, e.g., writing

\frac{x^2 + 4x}{x} = x^2 + 4

by 'crossing out' the

x

.

Correction: Factor the numerator first:

\frac{x(x+4)}{x} = x + 4

. You may only cancel a factor that multiplies the entire numerator with a factor that multiplies the entire denominator.

Concluding that two expressions are equivalent after checking only

x = 0

or

x = 1

, because many different expressions agree at these values.

Correction: Always test at a second, less convenient value such as

x = 3

or

x = -2

. Better yet, rely on full simplification and use substitution only to verify.

Forgetting to state restrictions when simplifying a rational expression, e.g., writing

\frac{x(x+2)}{x+2} = x

without noting

x \neq -2

.

Correction: Identify all values that make any denominator zero before cancelling, and carry those restrictions forward to your final answer.

Distributing a minus sign incorrectly, e.g., writing

-(x^2 - x + 4) = -x^2 - x + 4

instead of

-x^2 + x - 4

.

Correction: Multiply every term inside the bracket by

-1

:

-(x^2 - x + 4) = -x^2 + x - 4

.

Deciding two expressions are equivalent because their simplified polynomial forms match, without checking whether the original expressions have different restrictions.

Correction: Compare both the simplified form and the domain restrictions. Two expressions are fully equivalent only when both the simplified rule and the domain match.

Lesson summary

Two algebraic expressions are equivalent if they produce the same output for every value of the variable where both are defined.
The primary method is full simplification: expand, collect like terms, and for rational expressions factor and cancel. If the simplified forms are identical and the restrictions match, the expressions are equivalent.
Substitution is a fast tool for disproving equivalence — one counterexample is enough. However, a single matching substitution does not prove equivalence.
Avoid testing only at $x = 0$ or $x = 1$ ; these values often fail to reveal differences between non-equivalent expressions.
For rational expressions, always state restrictions (values excluded from the domain) and include them when deciding equivalence.
Cancel factors, never individual terms, when simplifying rational expressions.

Check your understanding

Question 1

Which statement best describes what it means for two algebraic expressions to be equivalent?

They produce the same output for at least one value of the variable.
They contain the same number of terms.
They produce the same output for every value of the variable where both are defined.
They look identical before any simplification.

Show answer and explanation

They produce the same output for every value of the variable where both are defined.

Equivalence requires matching outputs for every allowed input, not just one. Expressions can look different and still be equivalent, or look similar and not be equivalent.

Question 2

A student tests

x = 0

and

x = 1

and finds that expressions

A

and

B

give the same output both times. What can the student correctly conclude?

A and B are definitely equivalent.
A and B are not equivalent.
The test is inconclusive — more evidence, such as full simplification, is needed.
A and B are equivalent for all positive values of x.

Show answer and explanation

The test is inconclusive — more evidence, such as full simplification, is needed.

Two non-equivalent expressions can agree at x = 0 and x = 1. A substitution test can disprove equivalence but cannot prove it on its own. Full simplification is needed for a definitive answer.

Question 3

Simplify

\frac{x^2 - 9}{x + 3}

and choose the correct simplified form with its restriction.

$x - 3$ , no restrictions needed
$x + 3$ , $x \neq -3$
$x - 3$ , $x \neq -3$
$x^2 - 9$ , $x \neq -3$

Show answer and explanation

x - 3

,

x \neq -3

Factor the numerator as

(x-3)(x+3)

, then cancel the common factor

(x+3)

to get

x - 3

. The restriction

x \neq -3

must be stated because the original denominator is zero there.

Question 4

Are

M = 4(x + 2) - 3(x - 1)

and

N = x + 11

equivalent?

Yes, because both contain the variable x.
No, because M has brackets and N does not.
Yes, because both simplify to $x + 11$ .
No, because M simplifies to $x + 5$ .

Show answer and explanation

Yes, because both simplify to

x + 11

.

Expand M:

4x + 8 - 3x + 3 = x + 11

. This matches N exactly, so M and N are equivalent for all real values of x.

Key terms

Equivalent expressions: Two algebraic expressions that produce the same output for every value of the variable(s) where both expressions are defined.
Conditional equation: An equation that is true only for specific values of the variable, not for all values. For example, $x + 5 = 9$ is only true when $x = 4$ .
Rational expression: A fraction in which the numerator and denominator are both polynomials, such as $\frac{x^2 - 1}{x + 3}$ .
Restriction: A value of the variable that must be excluded from the domain because it makes a denominator equal to zero, making the expression undefined.
Counterexample: A single specific value of the variable for which two expressions give different outputs, proving the expressions are not equivalent.
Like terms: Terms that have exactly the same variable part (including exponent), such as $5x^2$ and $-2x^2$ . Only like terms can be added or subtracted directly.
Factoring: Rewriting an expression as a product of simpler expressions. For example, $x^2 - 4 = (x-2)(x+2)$ .
Domain: The set of all values of the variable for which an expression is defined (gives a real-number output).

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