Worksheet: Properties of Equality

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In this worksheet, we will practice dealing with the properties of equality and writing an equivalent expression to a given one.

Q1:

The two diagrams show 4 × 3 and 3 × 4 using an area model for multiplication.

What can be said about the areas of the rectangles?

  • AThey are not equal.
  • BThey are equal.

What does it say about multiplication?

  • AThe multiplication is associative: 4 × 3 = 3 × 4 .
  • BThe multiplication is commutative: 4 × 3 = 3 × 4 .

Q2:

Complete the equality stating which property is being used: 4 3 + = 9 + 4 3 .

  • A43, additive identity
  • B34, associative
  • C9, commutative
  • D52, associative

Q3:

What equivalent expression to 𝑎 + 𝑏 − 𝑐 does this diagram show?

  • A 𝑎 − 𝑏 + 𝑐
  • B 𝑏 − 𝑎 − 𝑐
  • C 𝑏 − 𝑎 + 𝑐
  • D 𝑏 + 𝑎 − 𝑐
  • E 𝑏 + 𝑎 + 𝑐

Q4:

The diagram shows a rectangular prism (A) that has been divided into three horizontal slices (B) and into four vertical slices (C). The volume of the rectangular prism as represented in (B) is written as 𝑉 = 3 × ( 𝑎 × 4 ) , where 1 × ( 𝑎 × 4 ) is the volume of one slice.

Choose an expression for 𝑉 as represented in (C).

  • A 𝑉 = ( 3 × 𝑎 ) + 3 × 4
  • B 𝑉 = ( 4 × 𝑎 ) + 4 × 3
  • C 𝑉 = ( 3 × 𝑎 ) + 4
  • D 𝑉 = ( 3 × 𝑎 ) × 4
  • E 𝑉 = ( 3 + 𝑎 ) + 4

What property of multiplication does this diagram show?

  • Athe commutative property
  • Bthe associative property
  • Cthe multiplicative inverse property
  • Dthe multiplicative identity property
  • Ethe distributive property

How many slices with a volume of 12 volume units are there in the rectangular prism?

  • A3
  • B 4 𝑎
  • C 3 𝑎
  • D4
  • E 𝑎

Q5:

Which of the following demonstrates the symmetric property of equality?

  • AFor any 𝑥 , 𝑥 = 𝑥 .
  • BIf 𝑥 = 𝑦 , then 𝑦 = 𝑥 .
  • CIf 𝑥 = 𝑦 and 𝑦 = 𝑧 , then 𝑥 = 𝑧 .

Q6:

Which of the following demonstrates the reflexive property of equality?

  • AFor any 𝑥 , 𝑥 = 𝑥 .
  • BIf 𝑥 = 𝑦 and 𝑦 = 𝑧 , then 𝑥 = 𝑧 .
  • CIf 𝑥 = 𝑦 , then 𝑦 = 𝑥 .

Q7:

Consider the following equality statements:

  1. 𝑥 + 3 = 7 𝑦 − 9 .
  2. 7 𝑦 − 9 = 𝑧 ( 𝑧 + 1 ) .
  3. 𝑦 = 4 𝑤 .

Without solving for 𝑥 , 𝑦 , 𝑧 , or 𝑤 , select the true statement.

  • A 𝑥 + 2 = 2 𝑥
  • B 𝑥 + 2 = 𝑥
  • C 2 𝑥 + 2 = 𝑥 + 2
  • D 𝑥 + 2 = 𝑥 − 2
  • E 𝑥 + 2 = 𝑥 + 2

Without solving for 𝑥 , 𝑦 , 𝑧 , or 𝑤 , select the true statement.

  • A 4 𝑤 = 𝑥 + 3
  • B 7 𝑦 − 9 = 𝑧 + 1
  • C 7 𝑦 − 9 = 4 𝑤
  • D 7 𝑦 − 9 = 𝑥 + 3
  • E 𝑧 + 1 = 𝑥 + 3

Without solving for 𝑥 , 𝑦 , 𝑧 , or 𝑤 , select the true statement.

  • A 4 𝑤 = 𝑧 ( 𝑧 + 1 )
  • B 4 𝑤 = 7 𝑦 + 9
  • C 𝑥 + 3 = 𝑧 + 1
  • D 4 𝑤 = 𝑥 + 3
  • E 𝑥 + 3 = 𝑧 ( 𝑧 + 1 )

Without solving for 𝑥 , 𝑦 , 𝑧 , or 𝑤 , select the true statement.

  • A 1 1 𝑤 − 9 = 𝑧 ( 𝑧 + 1 )
  • B 4 𝑤 − 9 = 𝑧 ( 𝑧 + 1 )
  • C 2 8 𝑤 − 9 = 𝑧 ( 𝑧 + 1 )
  • D 2 8 𝑤 − 9 = 𝑧 + 1
  • E 4 𝑤 − 9 = 𝑥 + 3

Q8:

Which of the following demonstrates the transitive property of equality?

  • AIf 𝑥 = 𝑦 and 𝑦 = 𝑧 , then 𝑥 = 𝑧 .
  • BIf 𝑥 = 𝑦 , then 𝑦 = 𝑥 .
  • CFor any 𝑥 , 𝑥 = 𝑥 .

Q9:

Which of the following figures has an area that can be represented by the expression 2 𝑥 + 8 𝑥 ?

  • A
  • B
  • C
  • D
  • E

Q10:

The diagram shows a rectangle of sides 𝑎 + 𝑏 and 𝑐 + 𝑑 . Its area is thus ( 𝑎 + 𝑏 ) × ( 𝑐 + 𝑑 ) , which can also be written as ( 𝑎 + 𝑏 ) ( 𝑐 + 𝑑 ) . Work out the area of the smaller rectangles that make up the bigger rectangle to find an equivalent expression to ( 𝑎 + 𝑏 ) ( 𝑐 + 𝑑 ) .

  • A 𝑎 + 𝑏 + 𝑐 + 𝑑
  • B 𝑎 𝑑 + 𝑏 𝑐
  • C 𝑎 𝑐 + 𝑎 𝑑 + 𝑏 𝑐 + 𝑏 𝑑
  • D 𝑎 𝑏 𝑐 𝑑
  • E 𝑎 𝑐 + 𝑏 𝑑

Q11:

Which of the following expressions is equivalent to 𝑧 + 𝑧 + 5 ?

  • A 2 𝑧 + 1
  • B 2 𝑧 + 5
  • C 5 𝑧 + 2
  • D 5 𝑧 + 1
  • E 𝑧 + 5

Q12:

Which of the following expressions is equivalent to 1 2 𝑡 + 1 4 𝑡 + 1 4 𝑡 − 3 ?

  • A 𝑡 − 3
  • B 3 𝑡 + 2
  • C 3 𝑡
  • D 2 𝑡
  • E 2 𝑡 − 3

Q13:

Which value of 𝑦 will make the following statement true: 𝑦 ( 2 𝑥 + 3 ) + 5 = − 8 𝑥 − 7 ?

Q14:

Which of the following expressions is equivalent to 𝑦 + 𝑦 + 𝑦 + 𝑦 ?

  • A 4 + 𝑦
  • B 5 + 𝑦
  • C 5 𝑦
  • D 4 𝑦
  • E 2 ( 𝑦 + 2 )

Q15:

Determine the value of 𝑦 that makes the equation 9 𝑥 + 1 1 𝑥 = 𝑦 𝑥 true for all values of 𝑥 .

Q16:

Determine the value of 𝑦 that makes the equation 7 ( 2 𝑥 ) = 𝑦 𝑥 true for all values of 𝑥 .

Q17:

Determine the value of 𝑦 that makes the equation 9 𝑥 + 4 + 7 𝑥 = 1 6 𝑥 + 𝑦 true for all values of 𝑥 .

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