Key Concepts

• Write equivalent expressions
• Analyze equivalent expressions
• Interpret equivalent expressions

4.8 Analyze equivalent expressions 

What is meant by equivalent expressions? 

Equivalent expressions are expressions that work the same even though they look different.  

If two algebraic expressions are equivalent, then the two expressions have the same value when we plug in the same value(s) for the variable(s). 

For example  

3(4x+6) and 12x+18 are the equivalent expressions. 

3(4x+6)  

Use the distributive property. 

3(4x+6)   

= 12x +18 

So, two equations are equivalent expressions. 

How to analyze equivalent expressions? 

• Two expressions are equivalent if they can be simplified to the same third expression or if one of the expressions can be written like the other.  
• In addition, you can also determine if two expressions are equivalent when values are substituted for the variable and both arrive at the same answer. 

4.8.1 Write equivalent expressions 

Example1: 

Horlicks claims that it contains 33% more than the usual pack. What expression shows the amount of Horlicks, h, in the new pack? 

Solution: 

Draw a bar diagram to represent the problem situation. Then write an expression to represent the amount of Horlicks in the new pack. 

h + 0.33h 

Combine like terms to write an equivalent expression. 

1(h) +0.33h 

=1.33h 

33 % more than 100% is the same as 133%. 

Example2: 

Find an equivalent expression for 12 x + 8. Show three more possible expressions. What do the rewritten expressions tell you about the relationship among the quantities? 

Solution: 

When you rewrite an expression, you are writing an equivalent expression. 

 12 x + 8 is equivalent to  

4(3x + 2) is equivalent to  

 3x + 3x+3x+3x + 2 +2+2+2. 

4.8.2 Analyze equivalent expressions 

Example1: 

Two groups went camp. They carry their water in reusable packs that come in three sizes. The table shows total packs every group carries. A medium water pack has 1 liter more than a small pack holds. A big pack holds 2 liters more than a small pack. Do the two groups carry the equal amount of water? If not, find group carries more? Use w to represent the number of liters of water a small pack can has. Show your work. 

Solution: 

First group: 

W +3(w+1) +2 (w+2) 

W + 3w+3+2w+4 

W + 3w +2w +3+4 

6w +7 

Second group: 

2w+w+1+3(w+2) 

2w+w+1+3w+6 

2w+w+3w+7 

6w+7 

Yes, they carry the same amount so two expressions are equivalent expressions. 

4.8.3 Interpret equivalent expressions  

Example1:

 

The total area, in square feet, of a rectangular mural that has been extended by x feet is represented by 5.5(7.5 + x). Expand the expression using the Distributive Property. What do each of the terms in the equivalent expression tell you about the mural? 

Solution: 

5.5 (7.5 + x) 

= (5.5 × 7.5) + (5.5 × x) 

= 41.25 +5.5x 

Example2: 

A rope is used to make a fence in the shape of an equilateral triangle around a newly planted tree. The length of the rope is represented with the expression 6x + 27.  

1. Rewrite the expression to represent the three side lengths of the rope fence.  

2. What is the length of one side? 

Solution: 

Length of the rope expression is 6x + 27. 

6x + 27 is equivalent to 3(2x+9) 

Three side length of the rope fence is 

3(2x+9) 

The length of the one side is 

 2x+9. 

Exercise:

1. Write equivalent expression to 3(x + 2) + 2x.

2. Which expressions are equivalent to 6a – a + 4b? Select all that apply.

5a + 4b

(6 – 1) a + 4b

(6 • 1) a + 4b

3. Are the expressions 3(x + y) + 2y + 10 and x + 5y + 2(x + 5) equivalent?Show your work.

4. The fine for an overdue library book is $0.55 for the first day and $0.50 for every additional day. Rewrite the expression 0.55 + 0.5(d – 1) for the fine as a sum of two terms where d represents the number of days overdue. Show your work.

5. A wading pool holds g gallons of water. A swimming pool holds 15 times as much water as the wading pool. Which expressions represent the total number of gallons of water in both pools?

6. Ria received a coupon for 15% off the total purchase price at a clothing store. Let c be the original price of the purchase. The expression c – 0.15c represents the new price of the purchase. Write an equivalent expression to show another way to represent the new price.

7. Find the expression equivalent to w+ 0.07w.

8. The perimeter of a square is represented with the expression 84+44s. What is the length of one side of the square?

9. John earns in dollars per hour. Next month he will receive a 3% raise in pay per hour. The expression n + 0.03n is one way to represent John’s pay per hour after the raise. Write an equivalent simplified expression that will represent his pay per hour after the raise.

10. Write as many expressions as you can that are equivalent to 10+24.

Concept Map

What have we learned:

• Analyse equivalent expressions
• Understand how to write equivalent expressions.
• Identify equivalent expressions.
• Interpret equivalent expressions.