Equivalent Expressions and Examples

Juwon says all three expressions are equivalent. 

1. Find the value of each expression for n=1 

2. Find the value of each expression for n=2.  

8n + 6, 2(4n + 3), 14n 

Solution: Substitute n=2, then find the values of the given expressions. 

3. Find the value of each expression for n=3.  

               8n + 6, 2(4n + 3), 14n 

Solution: Substitute n=2, then find the values of the given expressions.  

4. Critique Reasoning  

Do you agree with Juwon that all three expressions are equivalent? Explain.  

Solution:                     

No, I do not agree with Juwon. 

8n+6 and 2(4n+3) are equivalent. 

Because 2(4n+3) =2x4n + 2×3 = 8n+6 

We applied distributive property across addition 

a(b + c) = a(b) + a(c)  

When a number is substituted for the same variable in two expressions that are in the form of distributive property is always equal. 

Similarly, Properties of Operations  

1. Commutative property of addition and multiplication 

2. Associative property of addition and multiplication 

3. Distributive property across addition and subtraction             

Ok, let’s generate equivalent expressions. 

Equivalent expressions have the same value regardless of the value that is substituted for the same variable in the expressions.  

Properties of Operations  

Commutative Property  

           of addition a + b = b + a  

           of multiplication a x b = b x a  

Associative Property  

         of addition (a + b) + C = a + (b + c)  

         of multiplication (a x b) C = a × (b c)  

Distributive Property  

         across addition a(b + c) = a(b) + a(c)  

          across subtraction a(b – c) = a(b) – a(c) 

Example 1: 

Use properties of operations to write equivalent expressions for 3(4x – 1) and 2x + 4?  

Solution: 

Case1 

Use the distributive and associative properties to write an expression that is equivalent to 3(4x – 1). 

3(4x – 1) = 3(4x) – 3(1) ……. Distributive Property 

     = (3.4) x – 3  …..  Associative Property of multiplication 

∴ 3(4x – 1) = 12x – 3  

12x – 3 and 3(4x – 1) are equivalent expressions.  

Case2: 

Use the distributive property in reverse order to write an expression that is equivalent to 2x + 4.  

Look for a common factor of both terms that is greater than  

1. 2x + 4 =2(x) + 2(2) …………. Distributive property  

= 2(x+2) ……………2 is a common factor  

So, 2(x 2) is equivalent to 2x+4. 

Try It!  

Write an expression that is equivalent to 3x – 27. 

Solution: 

A common factor of 3 and 27 is 3 

3x – 27 = 3 (x) – 3 (9)   

=3(x– 9) 

So, 3x – 27 is equivalent to 3(y – 9). 

Example 2 

Which of the expressions below are equivalent?    

6x – 3            3x                  3(2x – 1) 

Solution: 

Step 1: 

Use the distributive property to simplify 3(2x – 1).  

3(2x – 1) = 3(2x) – 3(1)  

                  = 6x – 3  

So, 3(2x – 1) and 6x – 3 are equivalent expressions 

Step 2: 

Properties of operation cannot determine whether expressions are equivalent. 

Either   6x-3≠3x 

3(2x-1) ≠3x 

So, neither 6x-3 nor 3(2x-1) is equivalent to 4x. 

Example 3: 

Are 6(n+3) – 4 and 6n + 14 equivalent expressions? 

Solution: 

Step 1: 

Use properties of operations to simplify 6(n + 3) – 4.  

6(n + 3) – 4 = 6(n) + 6(3) – 4  ……………. Use the distributive property  

                        = 6n + 18 – 4    

6(n + 3) – 4 = 6n+14 

Generalize when two expressions name the same number regardless of the value of the variable, they are equivalent. 

Step 2: 

Substitute 3 for n to justify that the expressions are equivalent. 

Case 1: 

6(n + 3) – 4 = 6(3) + 6(3) – 4  

= 18 + 18 – 4                    

6(n + 3) – 4   = 32 

Case 2: 

6n+ 14   = (3) + 14  

                  = 18+ 14         

                       6n+ 14 = 32 

So, 6(n + 3) – 4 and 6n + 14 are equivalent expressions. 

Try It!  

Are 2(x – 3) + 1 and 2x + 6 equivalent expressions? Use substitution to justify your work. 

Solution: 

Use distributive property: 

 2(x – 3) + 1 = 2x – 6 +1 

                         = 2x – 5 

2(x – 3) + 1 = 2x – 5  

The given expression is 2x + 6 

So, 2x – 6 ≠ 2x + 6.                   

Practice and Problem solving. 

Critique Reasoning  

1. Jamie says that the expressions 6x – 2x + 4 and 4(x + 1) are not equivalent because one expression has a term that is subtracted and the other does not. Do you agree? Explain.  

Solution: 

Step 1: 

6x – 2x + 4 = 4x + 4  

Step 2: 

4(x + 1) = 4x + 4  

No, I don’t agree because the given expressions are equivalent.  

2. Are the two expressions shown below equivalent? Explain. 4(n + 3) – (3 + n) and 3n +9 

Solution: 

Step 1: 

Use distributive property: 

4(n + 3) – (3 + n) = 4n + 12 – 3 – n  

                                    = 4n – n + 12 – 3  

                                    = 3n + 9  

4(n + 3) – (3 + n) = 3n + 9 

So, 4(n+3) – (3+n) = 3n+9  

Higher Order Thinking  

3. Write an expression that has only one term and is equivalent to the expression below.  

(f.g² ) ÷ 5 – (g² . f)

Solution: 

(f.g² ) ÷5 – (g² . f) = f.g² / 5

=f.g² – 5 f.g²

= -4 f.g² / 5

Construct Arguments  

4. A Florida college golf team with 14 members is planning an awards banquet. To find the total cost of the meals, the team uses the expression 5 (g + 14), where g is the number of guests attending the banquet. A team member says that an equivalent expression is 5g + 14. Do you agree? Explain. 

Solution: 

I don’t agree with the team member, 

The equivalent fraction for 5(g + 14)  

5(g + 14) = 5(g) + 5(14) 

                    =5g + 70 

5g + 70 ≠ 5g + 14 

So, the statement given by the team member is wrong.  

Let’s Check Our Knowledge: 

1. Essential Question- How can you identify and write equivalent expressions?  

2. Use Structure- Which property of operations could you use to write an equivalent expression for y + 3? Write the equivalent expression.  

3. Generalize- Are z3 and 3z equivalent expressions? Explain. 

4. Are the expressions 3(y + 1) and 3y +3 equivalent for y=1? y = 27 y = 3?  

5. Construct Arguments- Are the expressions 3(y + 1) and 3y + 3 equivalent for any value of y? 

6. 6.Use properties of operations to complete the equivalent expressions. 

i). 3(x-6)  ii) 2x+10 iii) 8(2y + 2) iv) 5.7 +(3z +0.3)  

7. Write the letters of the expressions that are equivalent to the given expression. 

5(2x + 3)  

a. 10x + 15           b. 5x + 15x                c. 10x + 8 

8. Write an algebraic expression to represent the area of the rectangular rug. Then use properties of operations to write an equivalent expression.  

Answers: 

1. let us take the following expressions: 

 2x + 18 and 2(x+ 9) 

 2(x+9) = 2x + 18 

 So, 2x + 18 and 2(x+9) are equivalent expressions.  

 We can find that two expressions are equivalent by writing their simplest forms.  

2. By commutative property, a + b = b + a    then,  

 y+3 = 3+y 

3. The given expressions z3 and 3z are not equivalent expressions. 

z3 

≠ 3z  

4. The given expressions are 3(y + 1) and 3y +3  

If y = 1 

then, 3(y+1) = 3(1+1) 

                          = 3(2) 

           3(y+1) = 6 

           3y + 3   = 3(1) + 3 

           3y + 3   = 3 + 3 

           3y + 3  = 6 

the expressions 3(y+1) and 3y+3 are equivalent when substituting y=1 

If y = 27 

then, 

3(y+1) = 3(27+1) 

               = 3(28) 

3(y+1) = 84 

3y+3     = 3(27) + 3 

                = 81 + 3 

3y+3     = 84 

The expressions 3(y+1) and 3y+3 are equivalent when substituting y=27. 

If y = 3 

then, 

3(y+1) = 3(3+1) 

                = 3(4) 

3(y+1) = 12 

3y+3     = 3(3)+3 

                = 9+3 

3y+3     = 12 

The expressions 3(y+1) and 3y+3 are equivalent when substituting y=3. 

5. Given expressions 3(y+1) and 3y+3  

if y = 6 

then, 3(y+1) = 3(6+1) 

                          = 3(7) 

                          = 21 

3y+3     = 3(6)+3 

                = 18 + 3 

                = 21 

So, the given expressions are equivalent. 

6.  

1. 3 (x – 6) = 3x – 18  

2. 2x + 10 = 10 + 2x 

3. 8 (y + 2) = 8y + 16 

4. 5.7 + (3z + 0.3) = (5.7 + 3z) + 0.3 

7. Given expression is 5(2x – 3) 

5(2x – 3) = 10x – 15  

the option is “a.” 

8. Given, length of the rectangular rug = 2(x – 1) 

breadth of the rectangular rug = 5 

Area of the rectangular rug  

 = l× b sq. units 

 =2(x-1) × 5 

 = (2 × 5)(x – 1)……….. By commutative property of multiplication  

 = 10(x – 1) …………. By distributive property of subtraction.  

∴ Area of the rectangular rug = (10x – 10) sq. units 

Key Concept   

Two algebraic expressions are equivalent if they have the same value when any number is substituted for the variable. You can use the properties of operations to write equivalent expressions. 

Properties of Operations  

1. Commutative Property  

of addition a + b = b + a  

of multiplication a x b = b x a  

2. Associative Property  

of addition (a + b) + C = a + (b + c)  

of multiplication (a x b) C = a × (b c)  

3. Distributive Property  

across addition a(b + c) = a(b) + a(c)  

across subtraction a(b – c) = a(b) – a(c) 

Key concept covered : 

• Use properties of operations to write equivalent expressions. 
• Use properties of identify equivalent expressions. 
• Use substitution to justify equivalent expressions. 

Concept Map: