MODELING ALGEBRAIC EXPRESSIONS

Examples 1 -4 : Use a bar model to represent the following expressions.

Example 1 :

7 + x

Solution :

Example 2 :

z/3

Solution :

Example 3 :

t - 2

Solution :

Example 4 :

4y

Solution :

Example 5 :

Katrina and Andrew started the day with the same amount of money. Katrina spent 5 dollars on lunch. Andrew spent 3 dollars on lunch and 2 dollars on a snack after school. Do Katrina and Andrew have the same amount of money left ?

Solution :

Let x be the money that Katrina and Andrew had at the beginning. 

Step 1 :

Write an algebraic expression to represent the money Katrina has left.

x - 5

Represent the expression with a model.

Step 2 :

Write an algebraic expression to represent the money Andrew has left.

x - 3 - 2

Represent the expression with a model.

Step 3 :

Compare the models.

The models are equivalent, so the expressions are equivalent.

Andrew and Katrina have the same amount of money left.

Example 6 :

On a math quiz, Tina scored 3 points more than Julia. Juan scored 2 points more than Julia and earned 2 points in extra credit. Write an expression and draw a bar model to represent Tina’s score and Juan’s score. Did Tina and Juan make the same grade on the quiz ? Explain.

Solution :

Let y be the points scored by Julia.

Step 1 :

Write an algebraic expression to represent the points scored by Tina.

y + 3

Represent the expression with a model.

Step 2 :

Write an algebraic expression to represent the points scored by Juan.

y + 2 + 2

Represent the expression with a model.

Step 3 :

Compare the models.

The models are not equivalent, so the expressions are not equivalent.

Hence, Tina and Juan did not make the same grade on the quiz.

Example 7 :

Estaban is 6 years older than his brother Raoul. The sum of their ages is 38. How old are the brothers?

Solution :

Let x be the age of Raoul.

Age of Estaban = x + 6

The sum of the ages = 38

x + x + 6 = 38

2x + 6 = 38

2x = 38 - 6

2x = 32

x = 32/2

x = 16

Age of Raoul = 16 years

Age of Estaban = 16 + 6 ==> 22 years

Example 8 :

Two friends enter a trivia challenge as a team. Fayth scored 200 more points than Jamal. As a team, they collected a total of 2250 points. How many points did each friend earn?

Solution :

Let x be the points by Jamal.

Score of Fayth = x + 200

Total score = 2250

x + x + 200 = 2250

2x + 200 = 2250

2x = 2250 - 200

2x = 2050

x = 2050/2

x = 1025

Jamal's age = 1025

Fayath's age = 1025 + 200

= 1225

Example 9 :

Natalie, Chantal, and Samara play together as a forward line on a hockey team. At the end of the season, Chantal had scored eight more goals than Natalie while Samara had scored twice as many goals as Natalie. The three girls scored a total of 52 goals. How many goals did each girl score?

Solution :

Let x be the score of Natalie

Score by Chantal = x + 8

Score by Samara = 2x

Total score = 52

x + x + 8 + 2x = 52

4x + 8 = 52

4x = 52 - 8

4x = 44

x = 44/4

x = 11

x + 8 ==> 11 + 8 ==> 19

2x = 2(11) ==> 22

So, scores by Natalie, Chantal and Samara are 11, 19 and 22 respectively.

Example 10 :

The sum of three consecutive integers is 54. Find the numbers.

Solution :

Let x be the first number. Consecutive numbers are x + 1 and x + 2

Sum of these consecutive numbers = 54

x + x + 1 + x + 2 = 54

3x + 3 = 54

3x = 54 - 3

3x = 51

x = 51/3

x = 17

x + 1 ==> 17 + 1 ==> 18

x + 2 ==> 17 + 2 ==> 19

Example 11 :

Paloma works part-time, 4 hours per day, selling fitness club memberships. She is paid $9/h, plus a $12 commission for each 1-year membership she sells.

a) Write an algebraic expression that describes Paloma’s total earnings.

b) Find the amount Paloma makes in 8 hours when she sells 7 memberships.

Solution :

Let x be the number of membership she sells.

Number of hours she is working = 4

Earning per hour = $9

a) Total earning = 4(9) + 12x

= 36 + 12x

b)  Working for 8 hours, she is selling 7 memberships

= 8(9) + 12(7)

= 72 + 84

= 156

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