Evaluating Algebraic Expressions with Examples

Key Concepts

• To evaluate an algebraic expression, use substitution to replace the variable with a number.
• Evaluate algebraic expression with whole numbers
• Evaluate algebraic equations with decimals
• Evaluate algebraic expressions with fractions

Solve & Discuss It!  

A bike shop charges by the hour to rent a bike. Related items are rented for flat fees. Write an expression that represents how much it will cost to rent a bike and helmet for h hours. How much would it cost to rent a bike and a helmet for 3 hours?  

Expression for cost of renting a bike and helmet for h hours 

Rent for bike per hour = $12.50 

Rent for helmet per helmet = $5.25 

Rent for h hours = h x 12.50 + h x 5.25 

                                = h (12.50 + 5.25) 

Rent for 3 hours = 3(12.50 + 5.25) 

                                = 3(17.25) 

                                =$53.25 

Rent for bike and helmet for 3 hours is $53.25. 

Model with Math 

You can write an algebraic expression with decimals in the same way you do with whole numbers. 

Essential Question  

How can you evaluate an algebraic expression? 

Example 1: 

Erik collects miniature cars. He has one large case that has 20 cars.  

He also has 3 same-size, smaller cases filled with cars. 

Solution: 

Let n = the number of cars in each smaller case.  

How many miniature cars does Erik have if each  

smaller case holds 10 cars? 12 cars? 14 cars?  

To evaluate an algebraic expression, use substitution to replace the variable with a number. 

Evaluate 20 + 3n when n equals 10, 12 and 14. 

n = 10 

20 + 3n = 20 + 3(10)  

               =20 + 30 

               = 50 

If each smaller case holds 10 cars,  Erik has 50 cars. 

n = 12 

20 + 3n  = 20 + 3(12)  

                 =20 + 36 

                 =56  

If each smaller case holds 12 cars, Erik has 56 cars. 

n = 14  

20 + 3n = 20 + 3(14) 

                =20 + 42 

                =62 

If each smaller case holds 14 cars, Erik has 62 cars. 

Try it!  

Evaluate the expression 40 – t when t equals 10, 20, or 25. Then complete the table to show the values. 

Solution: 

Example 2: 

Julie’s family took a 4-day trip. Julie’s mother wrote an equation to calculate their gas mileage, m, in miles per gallon. Let d = the number of total miles driven on the trip. Let g = the total number of gallons of gas used for the trip.  

m = d/g. What was the gas mileage for the 4-day trip? 

Solution: 

Step 1  

Identify the values of the variables d and g. 

d = 476 + 439 + 382 + 263 = 1560 

g = 15 + 13.5 + 15.4 +16.1 = 60 

Step 2 

Substitute the values of the  

variables into the equation and  

evaluate. 

m = 1560/60 = 26  

The gas mileage was 26 miles per gallon. 

Try it!  

Evaluate the expression 3.4 + 12a – 4 for a = 10. 

Solution: 

Given expression is 3.4 +12a – 4  

The value of a = 10 

3.4 + 12a – 4 = 3.4 + 12 (10) – 4  

= 3.4 + 120 – 4  

= 123.4 – 4  

 = 119.4 

The value of the expression 3.4 + 12a – 4 when a = 10 is 119.4 

Example 2 

Mr. Grant wants to tile a 27-square-foot area with square tiles. Let s = the side length, in feet, of a square tile. Use the expression 27÷ s²

to find the number of tiles Mr. Grant needs to buy. 

Solution: 

Step 1: 

27 ÷ s² =

27÷ ( 1/3) ²

           = 27÷  ( 1/3. 1/3)

             = 27÷ 1/9

Step 2: 

Evaluate the expression 

27÷  1/9 = 27 ×  9/1

             = 243

Mr Grant needs to buy 243 tiles.  

Try it  

Suppose Mr. Grant decides to buy square tiles that have side lengths of 3/4 foot. How many tiles will he need to buy? 

Solution:

27 ÷ s² = 27÷ (3/4)²

= 27 ÷ (3/4 × 3/4) 

= 27 ÷ 9/16

Evaluate the expression 

27 ÷ 9/16 = 27× 16/9

= 48 

Mr. grant needs to buy 48 tiles if the side lengths are 3/4 foot. 

Key concept:

An algebraic expression can be written to represent a situation with an unknown quantity. Use a variable to represent the unknown quantity. An algebraic expression can be evaluated by substituting a value for the variable and performing the operations. 

Then use the order of operations to simplify 

Practice & Problem Solving 

1. The density, d, of an object can be found by using the formula d = m/v, where m is the mass of the object and visits volume. What is the density of an object that has a mass of 73,430 kilograms and a volume of 7 m³ ? 

Solution: 

Given that, 

Mass of the object = 73,430 kg 

Volume of the object = 7m³

Density = m/v

= 73,430 kg/ 7m³

= 10,490kg per m³

∴The density of the given object is 10,490 kg per m³

2. Tamara is making a medium-length necklace. Write an expression that shows how much it will cost Tamara for the chain pendant and b beads that cost $0.25 each. Then find the total cost of the necklace if Tamara uses 30 beads. 

Given that, 

Cost of each bead = $0.25 

No. of beads used = 30  

Total cost of medium necklace = $1.80 + 30 x $0.25 + $3.72 

                                                            = $1.80 + $7.5 + $3.72  

                                                            = $13.02 

∴ The total cost of medium necklace is $13.02. 

3. The formula V=s3 can be used to find the volume of a cube. Use the formula to find the volume, V, of a cube-shaped bin with side lengths of 2/3  yards. 

Solution: 

Given that, 

Length of the side of cube-shaped bin = 2/3 yards 

Volume of the cube = s³

= (2/3)³ 

= 2/3 x 2/3 x 23

= 8/27 cubic yards 

∴ The volume of the cube-shaped bin is 8/27 cubic yards 

Let’s check our knowledge: 

1. Evaluate each expression for t = 8, w = 1212 and x = 3 
   i. 3t – 8.              ii.   6w ÷ x + 9. 

2. Evaluate each expression for the value given.  
   i.  z÷4, z = 824. ii. 6÷9 – 22, t=60. 

3. Evaluate the given expression for x = 1.8, x = 5, and x = 6.4.  
   2x + 3.1  

4. Evaluate the given expression for the value given 8 – g ÷ 7878 when g = 5656 . 

5. Katie is evaluating the expression 15.75 ÷p+ 3p when p= 3.15. Explain each step that she should follow. 

Answers: 

1. Given that, 

               t = 8, w =1/2 and x = 3    

               i.   3t – 8 = 3 (8) – 8  

                        = 24 – 8  

                        = 16 

               ii.   6w ÷ x + 9 = 6(1/2) ÷ 3 + 9 

                             = 3 ÷ 3 + 9 

                             = 1 + 9 

                             = 10 

2.  
   i.   Given that, 
                  z = 824 
                  z ÷ 4 = 824 ÷ 4
                           = 206 
   
   ii.  Given that, 

                     t=60 

                    6t÷9 – 22 = 6(60) ÷ 9 – 22  

                                      = 360 ÷ 9 – 22  

                                      = 40 – 22  

                                      = 18 

3. Given that, 

               Case 1 

               x = 1.8 

               2x + 3.1 = 2(1.8) + 3.1 

                             =6.7 

               Case 2 

               x = 5 

               2x + 3.1 = 2(5) + 3.1 

                             = 10 + 3.1 

                            = 13.1 

               Case 3  

               x = 6.4 

               2x + 3.1 = 2(6.4) + 3.1 

                            = 15.9 

4. Given that, 

               g = 5/ 6

               8 – g ÷ 7/8 = 8 – 5/6 ÷ 7/8

                  = 8 – 5/6 x  8/7

                  = 8 – 20/21

                  = 168 −20/21

                  = 148/21

5. Given that, 

               p= 3.15 

               Step 1 

               15.75 ÷p+ 3p = 15.75 ÷ 3.15 + 3(3.15)   (Substitute the value of p in expression) 

               Step 2  

               15.75 ÷ 3.15 + 3(3.15) = 15.75 ÷ 3.15 + 9.45  (Simplify all terms) 

               Step 3  

               15.75 ÷ 3.15 + 9.45 = 5 + 9.45  (Use order of operations) 

                                        =14.45 

Exercise

1. Evaluate 30 + 4n when n equals 15, 16, or 17.
2. Evaluate the expression 20 – 3t when t equals 20, 26, or 34.
3. Mr. John wants to tile a 36 square foot area with square tiles. Let s= the side length in feet of a square tile. Use the expression 36+s² to find the number of tiles Mr. John needs to buy.
4. Evaluate the expression for t=8, w= ½ , and x=3.
   t² – 12w + x
5. Evaluate the expression for the value given.
   z÷4: z=824
6. Evaluate the expression for w=5, x=3, y=4, and z=8.
   W²+2÷48÷2z
7. Evaluate the expression for x=1.8, x= 5 and x=6.4.
   2x+3.1
8. Evaluate the expression for the value given.
   J+ ⅜ ; J =¾
9. Evaluate the expression for the value of b=8.9.
   b (3) + 20.4

Concept Map: