6th Grade Equivalent Expressions and the Distributive Property Worksheet Solutions (PDF)

6th Grade math worksheet solutions: Equivalent Expressions and the Distributive Property – Maths PDF

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This grade 6 equivalent expressions answers for 6th Grade covers equivalent expressions and the distributive property (Common Core 6.EE.A.3, 6.EE.A.4), with a free PDF to print.

Check each simplified expression here. The solutions show the distributive step, the like terms combined, and a check with a number when it helps. Area models and tables support several answers.

Grade 6 · Common Core State Standards: 6.EE.A.3, 6.EE.A.4

Grade 6 equivalent expressions answers: exercises 1 to 10

Exercise 1 – Distribute over a sum

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

$4x + 12$

The two parts have areas 4x and 12.

Exercise 2 – Distribute with a coefficient

$5 \cdot 2y = 10y$ and $5 \cdot 7 = 35$.

$10y + 35$

Exercise 3 – Distribute over a difference

$3 \cdot a = 3a$ and $3 \cdot 4 = 12$, with a minus sign between.

$3a – 12$

Exercise 4 – Distribute a variable sum

$6 \cdot m + 6 \cdot n$

$6m + 6n$

Exercise 5 – Factor out the GCF

The GCF of 8 and 12 is 4.

$8x = 4 \cdot 2x$ and $12 = 4 \cdot 3$.

$4(2x + 3)$

Rectangle of height 4.

Exercise 6 – Factor a sum

The GCF of 15 and 20 is 5.

$15 + 20y = 5 \cdot 3 + 5 \cdot 4y$

$5(3 + 4y)$

Exercise 7 – Combine like terms

The terms have the same variable part, so add the coefficients: $5 + 3 = 8$.

$8x$

Both expressions give the same values.

Exercise 8 – Combine several terms

Like terms: $7a + 3a = 10a$ and $2 + 9 = 11$.

$10a + 11$

Exercise 9 – Subtract like terms

$9y – 4y = 5y$.

$5y + 6$

Exercise 10 – Two variables

x-terms: $2x + 4x = 6x$. y-terms: $3y – y = 2y$.

$6x + 2y$

Exercises 11 and up

Exercise 11 – Which are equivalent?

$3(x + 2) = 3x + 6$ by the distributive property.

C is $3x + 6$. D is $x + x + x + 6 = 3x + 6$. B is missing the 3 times 2.

C and D

Exercise 12 – Test with a number

$2(4 + 5) = 2 \cdot 9 = 18$.

$2 \cdot 4 + 5 = 13$.

The values differ, so the expressions are not equivalent.

Not equivalent

Different values mean not equivalent.

Exercise 13 – Add versus multiply

$5 + 5 + 5 = 15$ and $3 \cdot 5 = 15$: equivalent.

$5 \cdot 5 = 25$, which is not 15.

$y + y + y = 3y$; $y \cdot y = y^{2}$ is different.

Exercise 14 – Area of a rectangle

Whole rectangle: $5(x + 2)$.

Two parts: $5x + 10$.

$5(x + 2) = 5x + 10$

Exercise 15 – Distribute then combine

$2(3x + 4) = 6x + 8$.

$6x + 8 + 5x = 11x + 8$.

$11x + 8$

Exercise 16 – Distribute a fraction

$\frac{1}{2} \cdot 8x = 4x$ and $\frac{1}{2} \cdot 6 = 3$.

$4x + 3$

Exercise 17 – Tickets and fees

$6t + 2t$ or $(6 + 2)t$.

$6t + 2t = 8t$.

For 5 tickets: $8 \cdot 5 = 40$.

$8t$; $40.00

Exercise 18 – Perimeter of a triangle

Add the sides: $(x + 3) + (x + 3) + (x + 3)$.

Or multiply: $3(x + 3) = 3x + 9$.

$3(x + 3) = 3x + 9$

Exercise 19 – Simplify with a difference

$10x – 3x = 7x$ and $2(x + 4) = 2x + 8$.

$7x + 2x + 8 = 9x + 8$

$9x + 8$

Exercise 20 – Multi-step expression

$3(2x + 5) = 6x + 15$.

$6x + 15 – 4x + 1 = 2x + 16$.

Check: $3(2 \cdot 2 + 5) – 4 \cdot 2 + 1 = 27 – 8 + 1 = 20$ and $2 \cdot 2 + 16 = 20$.

$2x + 16$

Key facts behind this grade 6 equivalent expressions answers

Key facts

Equivalent expressions have the same value for every value of the variable.
Distributive property: a(b + c) = ab + ac.
Like terms have the same variable part and can be combined.
Factor out the GCF to reverse the distributive property.
Substitute values to test whether two expressions are equivalent.

How to use this grade 6 equivalent expressions answers

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

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