Writing Linear Systems from Word Problems

Systems of Equations- Word Problems

Writing Equations

Objective: In this lesson you will learn how to read a systems of equations word problem in context and create two linear equations that represent the given situation.

​Step 1: Read the problem carefully

​Example problem: A drummer and a guitarist each wrote songs for their band. The guitarist wrote 8 fewer than the number of songs that the drummer wrote. They wrote a total of 46 songs.

Which system of equations models this situation if the drummer wrote d songs and the guitarist wrote g songs?

​I've highlighted two important sentences.

​Step 2: Analyze the sentences and create two linear equations.

​The guitarist wrote 8 fewer than the number of songs that the drummer wrote.

g = d - 8

They wrote a total of 46 songs.

g + d = 46

​Remember:

The drummer wrote d songs and the guitarist wrote g songs.

Multiple Choice

A drummer and a guitarist each wrote songs for their band. The guitarist wrote 8 fewer than the number of songs that the drummer wrote. They wrote a total of 46 songs.

Which system of equations models this situation if the drummer wrote d songs and the guitarist wrote g songs?

• $$g=d-8$$  

  $$g+d=46$$  

• $$g=8-d$$

  $$g=46-d$$

• $$d=g-8$$

  $$d=46-g$$

• $$d=8-g$$

  $$d+g=46$$

Match

Match the following equations to their situations:

Let x represent the number of cookies

Let y represent the number of brownies

• y = 2x

• x + y = 15

• y - x = 15

• y = 1/2x

• y = 2x + 15

There were twice as many brownies as cookies.

There was a total of 15 brownies and cookies.

The difference between the brownies and cookies was 15.

There were half as many brownies as cookies.

The brownies equaled 15 more than twice the number of cookies

• There were twice as many brownies as cookies.

• There was a total of 15 brownies and cookies.

• The difference between the brownies and cookies was 15.

• There were half as many brownies as cookies.

• The brownies equaled 15 more than twice the number of cookies

Multiple Choice

On Monday Joe bought 10 cups of coffee and 5 doughnuts for his office at the cost of $16.50.  The doughnuts were more popular than the coffee, so on Tuesday he bought 5 cups of coffee and 10 doughnuts for a total of $14.25. 

Which system of equations could be used to represent this situation? 

• 10c + 5d = 14.25
  5c + 10d = 16.50
• 10c + 5d = 16.50
  5c + 10d = 14.25
• c + d = 10
  5c + 10d = 16.50
• c + d = 5
  5c + 10d = 16.50

Multiple Choice

Last season two running backs on the Steelers football team rushed a combined total of 1550 yards. One rushed 4 times as many yards as the other. Let x and y represent the number of yards each individual player rushed.

Which system of equations could be used to represent this situation?

• x + y = 1550

  y = 4x

• x + y = 1550

  y = x + 4

• y - x = 1550

  y = 4x

• y = 1550 + x

  y = x + 4

Multiple Choice

Derek mowed his next door neighbor’s lawn for a handful of dimes and nickels, 80 coins in all.  Upon completing the job he counted out the coins, and the total came to $6.60. 

Which system of equations could be used to find the exact number of dimes and nickels? 

• d + n = 6.60
  0.10d + 0.05n = 80

• d + n = 80
  10d + 5n = 6.60

• d + n = 80
  0.10d + 0.05n = 6.60

• d + n = 80
  0.05d + 0.10n = 6.60

Multiple Choice

Stefan's school is selling tickets to a musical performance. On the first day of sales, the school sold 3 senior citizen tickets and 1 child ticket for a total of $38. The school took in $52 on the second day by selling 3 senior citizen tickets and 2 child tickets.

Which equations represents the system that could be used?

• 1s + 3c = 38

  2s + 3c = 52

• 3s + 1c = 38

  3s + 2c = 52

• s + c = 38

  s + c = 52

• 3s + 3c = 38

  1s + 2c = 52

Multiple Choice

A large pizza at Palanzio’s Pizzeria costs $6.80 plus 90 cents for each topping. The cost of a large pizza at Guido’s Pizza is $7.30 plus 65 cents for each topping. 

Which system of equations could be used to find the number of toppings when both companies cost the same amount? 

• y = 6.80 + 0.65x
  y = 7.30 + 0.90x

• x + y = 6.80
  x + y = 7.30
• y = 6.80 + 0.90x
  y = 7.30 + 0.65x

• y = 6.80 + 90x

  y = 7.30 + 65x