Identify Equivalent Expressions - 6th Grade Math

1. Identify whether $5(x-4)$ and $5x-4$ are equivalent.

   No, $5(x-4)$ is equivalent to $5x-20$, not $5x-4$. Distribute: $5(x-4) = 5x - 20$.

2. Which expression is equivalent to $\frac{3}{4}y+\frac{1}{4}y$?

   $y$. Sum the fractions: $\frac{3}{4}+\frac{1}{4}=\frac{4}{4}=1$.

3. What is the simplified form of $x+x+x+x+x$?

   $5x$. Five $x$'s added together equals $5x$.

4. Identify whether $x+x^2$ and $2x^2$ are equivalent.

   No, $x+x^2$ is not equivalent to $2x^2$. Can't combine unlike terms ($x$ and $x^2$).

5. Which expression is equivalent to $2a+3b+a-2b$?

   $3a+b$. Group like terms: $(2a + a) + (3b - 2b) = 3a + b$.

6. Identify whether $y+y+y$ and $3y$ are equivalent.

   Yes, $y+y+y$ is equivalent to $3y$. Three $y$'s added together equals $3$ times $y$.

7. Which expression is equivalent to $8-2(3x-1)$?

   $10-6x$. Distribute $-2$: $8 - 6x + 2 = 10 - 6x$.

8. Identify whether $\frac{x}{3}+\frac{x}{3}+\frac{x}{3}$ and $x$ are equivalent.

   Yes, $\frac{x}{3}+\frac{x}{3}+\frac{x}{3}$ is equivalent to $x$. Three thirds make a whole: $\frac{3}{3}x = x$.

9. Which property justifies rewriting $a+a+a$ as $3a$?

   Combining like terms (repeated addition as multiplication). Adding the same term multiple times equals multiplying by the count.

10. Which expression is equivalent to $6(2x+1)$?

    $12x+6$. Distribute $6$ to both terms: $6 \cdot 2x + 6 \cdot 1$.

11. Identify the expression equivalent to $6(p+1)-p$ after simplifying.

    $5p+6$. Distribute $6$, then combine: $6p+6-p=5p+6$.

12. Which expression is equivalent to $7y+0$?

    $7y$. Adding zero doesn't change the value.

13. Which expression is equivalent to $9-3x+2x$?

    $9-x$. Combine like terms: $-3x + 2x = -x$, so $9 - x$.

14. Which expression is equivalent to $\frac{3}{4}y+\frac{1}{4}y$?

    $y$. Add fractions: $\frac{3}{4} + \frac{1}{4} = 1$, so $1y = y$.

15. Identify the expression equivalent to $(x+3)+(2x-1)$ after combining like terms.

    $3x+2$. Combine $x$ terms ($x+2x=3x$) and constants ($3-1=2$).

16. Which expression is equivalent to $7a-a$?

    $6a$. Subtracting $a$ from $7a$ leaves $6a$.

17. Which value of $x$ shows $x+5$ and $5x$ are not equivalent?

    $x=0$. When $x=0$: $0+5=5$ but $5(0)=0$, showing they differ.

18. Which expression is equivalent to $5(x-4)$?

    $5x-20$. Distribute $5$ to both terms: $5 \cdot x$ and $5 \cdot (-4)$.

19. Which expression is equivalent to $2(x-5)$?

    $2x-10$. Distribute $2$ to both terms: $2 \cdot x - 2 \cdot 5$.

20. Find the value of $k$ so $k(x+1)$ is equivalent to $4x+4$.

    $k=4$. Expanding gives $kx+k$; matching coefficients: $k=4$.

21. What is the simplified form of $6(2+p)$?

    $12+6p$. Distribute: $6 cdot 2 + 6 cdot p = 12 + 6p$.

22. Which expression is equivalent to $y+y+y$?

    $3y$. Adding $y$ three times equals $3$ times $y$.

23. Which expression is equivalent to $x+x+x+x+x$?

    $5x$. Adding $x$ five times equals $5$ times $x$.

24. Which expression is equivalent to $4x+2x$?

    $6x$. Combine like terms: $4x + 2x = (4+2)x = 6x$.

25. Which expression is equivalent to $10-(x+2)$?

    $8-x$. Distribute the negative: $10 - x - 2 = 8 - x$.

26. Which expression is equivalent to $12m+4-3m$?

    $9m+4$. Combine like terms: $12m - 3m = 9m$.

27. Identify the expression equivalent to $10-3(x-2)$ after simplifying.

    $16-3x$. Distribute the $-3$: $10-3x+6=16-3x$.

28. What is the simplified form of $3(4-t)$?

    $12-3t$. Distribute: $3 cdot 4 - 3 cdot t = 12 - 3t$.

29. What does it mean for two expressions to be equivalent?

    They have the same value for every allowed variable value. Equivalent expressions produce identical outputs for all valid inputs.

30. Identify whether $x+5+x$ and $2x+5$ are equivalent.

    Equivalent. Rearranging gives $x + x + 5 = 2x + 5$.

31. Which expression is equivalent to $4x+9-2x$?

    $2x+9$. Combine like terms: $4x-2x=2x$, constant stays $9$.

32. Which expression is equivalent to $2(x+5)+3(x-1)$?

    $5x+7$. Expand both: $2x+10+3x-3$, then combine like terms.

33. Find and correct the error: $4(x+1)=4x+1$.

    Correct: $4(x+1)=4x+4$. Must distribute $4$ to both terms inside parentheses.

34. Identify the simplified expression equivalent to $7x-x$.

    $6x$. Subtracting one $x$ from seven $x$'s leaves six $x$'s.

35. Which is equivalent to $2(x+y)$ using the distributive property?

    $2x+2y$. Multiply $2$ by each term inside the parentheses.

36. Identify whether $2x+3$ and $2(x+3)$ are equivalent.

    Not equivalent. $2(x+3)=2x+6$, which differs from $2x+3$.

37. Identify whether $3x+6$ and $3(x+2)$ are equivalent.

    Equivalent. $3(x+2)=3x+6$, which matches the first expression.

38. Which expression is equivalent to $\frac{1}{2}x+\frac{1}{2}x$?

    $x$. Add fractions: $\frac{1}{2} + \frac{1}{2} = 1$, so $1x = x$.

39. What is the simplified form of $2x+3x$?

    $5x$. Combine like terms: $(2+3)x = 5x$.

40. Identify the missing term so $x+\square+x$ is equivalent to $3x$.

    $x$. Need three $x$'s total, so add one more $x$.

41. Which expression is equivalent to $3(n+2)$?

    $3n+6$. Distribute $3$ to both terms inside the parentheses.

42. Identify whether $2(x+5)+3x$ and $5x+10$ are equivalent.

    Yes, $2(x+5)+3x$ simplifies to $5x+10$. Distribute first: $2x+10+3x = 5x+10$.

43. Which expression is equivalent to $5(x+1)+2x$?

    $7x+5$. Distribute $5$, then combine like terms: $5x + 5 + 2x = 7x + 5$.

44. Identify whether $3(x+2)$ and $3x+6$ are equivalent.

    Yes, $3(x+2)$ is equivalent to $3x+6$. Distribute: $3(x+2) = 3x + 6$.

45. Which expression is equivalent to $2(x+4)+3(x+4)$?

    $5(x+4)$. Factor out $(x+4)$: $(2+3)(x+4) = 5(x+4)$.

46. Identify whether $3(x-4)$ and $3x-12$ are equivalent.

    Equivalent. Distributing $3$ gives $3x - 12$, matching the second expression.

47. Identify whether $2(x+3)$ and $2x+3$ are equivalent.

    No, $2(x+3)$ is equivalent to $2x+6$, not $2x+3$. Distribute: $2(x+3) = 2 cdot x + 2 cdot 3 = 2x+6$.

48. What is the simplified form of $4a-2a$?

    $2a$. Combine like terms: $4a-2a = (4-2)a = 2a$.

49. What is the simplified form of $9m-m$?

    $8m$. Subtract: $9m - 1m = 8m$.

50. Identify whether $x+2y$ and $2y+x$ are equivalent.

    Yes, $x+2y$ is equivalent to $2y+x$. Commutative property allows reordering addition.

51. Identify the simplified expression equivalent to $y+y+y+y$.

    $4y$. Four $y$'s added together equals $4$ times $y$.

52. Which expression is equivalent to $1\text{ }x$?

    $x$. Multiplying by $1$ doesn't change the value.

53. Identify the expression equivalent to $2a+3b-a+b$ after combining like terms.

    $a+4b$. Group like terms: $2a-a=a$ and $3b+b=4b$.

54. Which expression is equivalent to $3(b+2)$?

    $3b+6$. Distribute $3$ to both terms: $3 \cdot b + 3 \cdot 2$.

55. Identify the simplified expression equivalent to $\frac{1}{2}x+\frac{1}{2}x$.

    $x$. Adding two halves gives one whole: $\frac{1}{2}+\frac{1}{2}=1$.

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