Identifying When Expressions Are Equivalent

In this lesson:

• Define what expression equivalence means
• Test expressions using substitution
• Verify equivalence with algebra

Learning Objectives for This Lesson

By the end of this lesson, you should be able to:

1. State what equivalence means — same result for every value substituted
2. Use substitution with multiple values to test expressions
3. Verify equivalence algebraically using properties of operations

What Do You Already Know?

You can already:

• Substitute values into expressions and evaluate them
• Use the distributive property:
• Combine like terms:

Today's question: When are two expressions always equal — not just once?

Two Expressions Name the Same Value

Equivalence means: both expressions give the same result for every value of the variable.

• Not just for one value — for all values
• A universal statement, not a single check

Equivalent expressions name the same number for any substituted value.

The Standard Example: Three Values of

Both columns agree for every value tested.

Why Substitution Is Not Enough

Substitution can disprove equivalence with one mismatch.

Substitution cannot prove equivalence — there are infinitely many values to check.

We need algebraic verification to be certain.

Algebraic Proof for the Standard Example

We use the distributive property in reverse — combining like terms.

Both expressions simplify to the same form: . They are equivalent. ✓

Two Steps to Classify Any Pair

Step 1 — Test with substitution:

• Substitute 2–3 values into both expressions
• One mismatch → not equivalent (stop here)
• All match → proceed to Step 2

Step 2 — Verify algebraically:

• Simplify using properties of operations
• Both simplify to same form → equivalent ✓

Example 1: Are and Equivalent?

Step 1 — Test: Let

Step 2 — Verify: Distribute

Both simplify to . Equivalent. ✓

Example 2: Are and Equivalent?

Step 1 — Test: Let

— Counterexample found. Not equivalent. ✗

No algebraic verification needed — one mismatch is enough.

Quick Check: Predict Before You Compute

Are and equivalent?

Think: what does the distributive property say?

Substitute to test — then check the next slide.

Example 3: Are and Equivalent?

Step 1 — Test: Let

Step 2 — Verify: Distribute

Equivalent. ✓

Example 4: Different Look, Same Value

Are and equivalent?

Test: Let

Verify: ✓ Equivalent.

Quick Check: Similar but Different?

Are and equivalent?

They both contain 5, n, and 3 — does that mean they're the same?

Pick a value of and test. What do you find?

Example 5: Similar-Looking but Not Equivalent

Are and equivalent?

Test: Let

— Not equivalent. ✗

, not .

Watch Out: One Match Is Not Enough

Consider and :

• At : both equal 12 ✓
• At : ✗

One agreement does not prove equivalence.

Finding a Counterexample: vs.

One counterexample is sufficient to disprove.

Distributing to Every Term Matters

Common error:

Correct: Draw the arrow to both terms:

Incomplete distribution leads to wrong equivalence judgments.

Looking Similar Does Not Mean Equivalent

Never judge by appearance — always test and verify.

Practice: Classify These Three Pairs

For each pair: substitute a value, then verify or find the counterexample.

Pair A: and

Pair B: and

Pair C: and

Work each one, then advance for answers.

Practice Answers: Three Pairs Classified

Pair A: and → Equivalent ✓

• Distribute:

Pair B: and → Not equivalent ✗

Pair C: and → Equivalent ✓

• Combine like terms:

What to Remember from This Lesson

✓ Equivalence means same result for every value — not just one

✓ Test: one mismatch disproves immediately

✓ Verify algebraically — substitution tests, algebra proves

One match does not prove equivalence

Distribute to every term inside parentheses

Looking similar ≠ equivalent

What Comes Next: Equations and Solutions

Today: Two expressions are equivalent — always equal, for every value.

Next (6.EE.B.5): An equation asks for which specific value makes two expressions equal — not always, just at one point.

• is true only when (a solution, not equivalence)