Identify When Two Expressions Are Equivalent

$x + 12$

$3x + 3$

$2x + 4$

$4x + 1$

They look the same when written.

They give the same result for one specific value.

They give the same result for every value substituted.

They have the same number of terms.

$2n + 3$

$n + 6$

$2n + 6$

$2n + 5$

No — they give different results.

Yes — $y + y + y = 15$ and $3y = 15$, so they match at $y = 5$.

Cannot be determined from one test.

Yes — because both expressions have $y$ in them.

Both equal 8 at $n = 1$, so they might be equivalent.

Both equal 20 at $n = 1$, so they are equivalent.

$5(1) + 3 = 8$ and $5(1 + 3) = 20$. They differ, so the expressions are NOT equivalent.

They are equivalent because both contain 5 and 3.

Yes — both expressions have 3, $x$, and 2.

Yes — $3(x + 2) = 3x + 2$ by the distributive property.

No — $3(x + 2) = 3x + 6$ by the distributive property. $3x + 2 \neq 3x + 6$.

Cannot be determined without knowing the value of $x$.

$4y + 2$ and $2(2y + 2)$

$3a + 6$ and $3(a + 2)$

$2x + 4$ and $2(x + 4)$

$5n + 3$ and $5(n + 3)$

They need to test more values before deciding.

The expressions are equivalent because they look similar.

One counterexample is enough — the expressions are NOT equivalent.

They need to try $x = 0$ specifically.

At $x = 6$: $x + 6 = 12$ and $2x = 12$. These are equal, so they ARE equivalent.

At $x = 4$: $x + 6 = 10$ and $2x = 8$. These differ, so they are NOT equivalent.

The expressions look different, so they cannot be equivalent.

At $x = 6$: $x + 6 = 12$ and $2x = 12$. These are equal, so no counterexample exists.

Testing three values and getting the same result each time.

Showing the expressions have the same number of terms.

Simplifying both expressions to the same form using algebraic properties.

Testing one value and getting the same result.

$y + y + y$ and $3y$

$2(n + 3)$ and $2n + 6$

$4y + 2$ and $2(2y + 2)$

$3(a + 2)$ and $3a + 6$

Yes — both describe a perimeter, so they must be equal.

No — Student A wrote addition and Student B did not include parentheses.

Yes — $2(x + 2) + 2(3) = 2x + 4 + 6 = 2x + 10$, which equals Student B's expression.

Cannot be determined without a value for $x$.

He calculated $x + 6$ incorrectly.

He tested the wrong value of $x$.

He concluded equivalence from one matching substitution. One test is not enough — he should also verify algebraically or test another value (e.g., $x = 5$ gives $11 \neq 10$).

The expressions are actually equivalent — Jordan is correct.

She used an incorrect value of $x$.

She incorrectly simplified $3(x + 2)$ as $3x + 2$ instead of $3x + 6$. Her test compared $3x + 2$ to $3(x + 2)$, not $3x + 6$ to $3(x + 2)$. The expressions $3(x + 2)$ and $3x + 6$ ARE equivalent.

She should have tested more values before concluding.

Her conclusion is correct — $3(x + 2)$ and $3x + 6$ are not equivalent.