Worksheet: Properties of Equality

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In this worksheet, we will practice defining the properties of equality and using them to write 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 equal.
  • BThey are not 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: 43+=9+43.

  • A43, additive identity
  • B52, associative
  • C9, commutative
  • D34, associative

Q3:

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

  • A𝑎−𝑏+𝑐
  • B𝑏−𝑎−𝑐
  • C𝑏+𝑎−𝑐
  • D𝑏−𝑎+𝑐
  • E𝑏+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).

  • A𝑉=(3×𝑎)+3×4
  • B𝑉=(3+𝑎)+4
  • C𝑉=(3×𝑎)×4
  • D𝑉=(3×𝑎)+4
  • E𝑉=(4×𝑎)+4×3

What property of multiplication does this diagram show?

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

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

  • A4
  • B𝑎
  • C3𝑎
  • D3
  • E4𝑎

Q5:

Which of the following demonstrates the symmetric property of equality?

  • AIf 𝑥=𝑦, then 𝑦=𝑥.
  • BFor any 𝑥, 𝑥=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, then 𝑦=𝑥.
  • CIf 𝑥=𝑦 and 𝑦=𝑧, then 𝑥=𝑧.

Q7:

Consider the following equality statements:

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

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

  • A𝑥+2=𝑥+2
  • B𝑥+2=𝑥
  • C2𝑥+2=𝑥+2
  • D𝑥+2=𝑥−2
  • E𝑥+2=2𝑥

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

  • A7𝑦−9=𝑧+1
  • B7𝑦−9=4𝑤
  • C4𝑤=𝑥+3
  • D7𝑦−9=𝑥+3
  • E𝑧+1=𝑥+3

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

  • A4𝑤=𝑧(𝑧+1)
  • B4𝑤=7𝑦+9
  • C𝑥+3=𝑧(𝑧+1)
  • D𝑥+3=𝑧+1
  • E4𝑤=𝑥+3

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

  • A4𝑤−9=𝑧(𝑧+1)
  • B4𝑤−9=𝑥+3
  • C11𝑤−9=𝑧(𝑧+1)
  • D28𝑤−9=𝑧+1
  • E28𝑤−9=𝑧(𝑧+1)

Q8:

Which of the following demonstrates the transitive property of equality?

  • AIf 𝑥=𝑦 and 𝑦=𝑧, then 𝑥=𝑧.
  • BFor any 𝑥, 𝑥=𝑥.
  • CIf 𝑥=𝑦, then 𝑦=𝑥.

Q9:

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

  • 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).

  • A𝑎𝑑+𝑏𝑐
  • B𝑎𝑏𝑐𝑑
  • C𝑎𝑐+𝑏𝑑
  • D𝑎+𝑏+𝑐+𝑑
  • E𝑎𝑐+𝑎𝑑+𝑏𝑐+𝑏𝑑

Q11:

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

  • A2𝑧+1
  • B𝑧+5
  • C5𝑧+2
  • D2𝑧+5
  • E5𝑧+1

Q12:

Which of the following expressions is equivalent to 12𝑡+14𝑡+14𝑡−3?

  • A2𝑡
  • B2𝑡−3
  • C3𝑡+2
  • D𝑡−3
  • E3𝑡

Q13:

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

Q14:

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

  • A4+𝑦
  • B5𝑦
  • C5+𝑦
  • D2(𝑦+2)
  • E4𝑦

Q15:

Determine the value of 𝑦 that makes the equation 9𝑥+11𝑥=𝑦𝑥 true for all values of 𝑥.

Q16:

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

Q17:

Determine the value of 𝑦 that makes the equation 9𝑥+4+7𝑥=16𝑥+𝑦 true for all values of 𝑥.

Q18:

If 𝑎+𝑏=𝑏+𝑐, then 𝑐=.

  • A𝑎
  • B𝑎+𝑏
  • C0
  • D𝑏

Q19:

If the equation 2𝑥−4=2 has the same solution set as the equation 𝑎𝑥+18=21, what is the value of 𝑎?

Q20:

William believes the equation 𝑎𝑥+5=𝑏(2𝑥+2) can have more than one solution for special values of 𝑎 and 𝑏. Is he right? If so, what are the values?

  • Ayes, 𝑎=52, 𝑏=5
  • Bno
  • Cyes, 𝑎=5, 𝑏=52
  • Dyes, 𝑎=0, 𝑏=52

Q21:

Determine the value of 𝑦 that makes the equation 6(𝑥+8)+9𝑥=𝑦𝑥+48 true for all values of 𝑥.

Q22:

Given that 16𝑎𝑏+26𝑎=2𝑎8𝑎𝑏+𝑘, what is the value of 𝑘?

Q23:

Given that 2𝑥+9𝑥−23𝑥−30=𝑎𝑥+𝑏𝑥+𝑐(𝑥+6), by comparing coefficients, find 𝑎,𝑏, and 𝑐.

  • A𝑎=2, 𝑏=−3, 𝑐=−5
  • B𝑎=2, 𝑏=−3, 𝑐=5
  • C𝑎=2, 𝑏=21, 𝑐=−5
  • D𝑎=2, 𝑏=−3, 𝑐=−30
  • E𝑎=2, 𝑏=9, 𝑐=−23

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