System of Equation Word Problems

Aim: Solving System of Equation Word Problems

by Ms. Medici & Mr. Jorgensen

​DO NOW: What is the solution to the following two linear equations shown on the graph?

​Lizzy has 30 coins that total $4.80. All of her coins are dimes, D, and quarters, Q. Write a system of equations to model this situation?

​Step #1: Identify the 2 totals.

​

​You have learned many different strategies for solving systems of equations! These include Graphing Systems of Equation, Substitution Method, Elimination, Addition Method, and my favorite using the calculator!

Now we are ready to apply these strategies to solve real world problems!

​Lets look at an Example:

​Mirna sold food at a soccer game to earn some extra money. She sold Pupusas for $2.00 and charamusca for $1.00, At the end of the match she made 15 dollars. She sold 10 pupusas and charamuscas combined. How many pupusas were sold and how many charamuscas were sold?

Lets start by identifying the important information:

• ​Pupusas cost $2

• ​Marachusca cost $1

• ​She sold 10 pupusas and marachuscas combined

• ​She made a total of $15

Step 1: Identify Totals:

Total money made=$15

Total Number of items sold =10​

Step 2: Define the variables:

In this problem we don't know how many pupusas were sold and how many marachuscas so these are our variables. Let P represent the number of pupusas sold and M represent the number of Marachuscas sold.

​Step 3: Determine the cost of each variable:

​P=$2

​M=$1

​Step 4: Write the first Equation: This first equation can represent the total number of food items sold.

​P + M =10

​Step 5: Write the second Equation: This second equation can represent the total number of money made.

​2P + 1M= $15

​Step 6: Write the equations so that the second is directly under the first and solve using your preferred method:

​P+M=10 P=10-M

​2P+1M=$15 2(10-M) + 1M=$15

​20-2M +1M =$15 -1M=-5 M=5 P+5= 10 P=5

​Lizzy has 30 coins that total $4.80. All of her coins are dimes, D, and quarters, Q. Write a system of equations to model this situation?

• ​Step #1: Identify the 2 totals:

​In this case, we know the total number of coins and total cost.

Total number of Coins = 30

Total Cost = $4.80

• ​Step #2: Identify the variables.

What are the two items that you are comparing or trying to find?

Dimes = D

Quarters = Q

• ​Step #3: Determine the cost or amount of each item.

Since Dimes cost 25 cents and Quarters cost 25 cents, we can say:

Dimes = $0.10

Quarters = $0.25

• ​Step #4: Write the first equation.

We will start by writing an equation to show there are 30 coins. 

Since Lizzy only has dimes and quarters, our first equation will look like this:

D + Q = 30

• Step#5: Write the second equation.

We will then write an equation to show the total cost is $4.80. 

Since the cost of each dime is $0.10 and the cost of each quarter is $0.25, our second equation will look like this:

0.10D + 0.25Q = $4.80

​

• Step#6: Rewrite the equations so that the second equation is directly under the first equation.

D + Q = 30

0.10D + 0.25Q = $4.80

Therefore, the system of equations for this scenario is:

D + Q = 30

0.10D + 0.25Q = 4.80.

​

Multiple Choice

Identify the 2 "totals" in the following word problem:

The Cinema sold a total of 150 tickets to a movie. Some of these were child tickets and the rest were adult tickets. A child ticket cost $7.75 and an adult ticket cost $10.25. If the cinema sold a total of $1470 worth of tickets, determine how many child tickets, c, and how many adult tickets, a, were sold.

• $7.75 & $10.25

• 150 tickets & $1470

• $7.75 & 150 tickets

Open Ended

Step #2 Identify the "variables" or the 2 items that you are comparing or trying to find:

The Cinema sold a total of 150 tickets to a movie. Some of these were child tickets and the rest were adult tickets. A child ticket cost $7.75 and an adult ticket cost $10.25. If the cinema sold a total of $1470 worth of tickets, determine how many child tickets, c, and how many adult tickets, a, were sold.

Open Ended

Step #3 Determine the cost of a child ticket and adult ticket:

The Cinema sold a total of 150 tickets to a movie. Some of these were child tickets and the rest were adult tickets. A child ticket cost $7.75 and an adult ticket cost $10.25. If the cinema sold a total of $1470 worth of tickets, determine how many child tickets, c, and how many adult tickets, a, were sold.

Multiple Choice

Which equation would represent the amount of tickets sold?

The Cinema sold a total of 150 tickets to a movie. Some of these were child tickets and the rest were adult tickets. A child ticket cost $7.75 and an adult ticket cost $10.25. If the cinema sold a total of $1470 worth of tickets, determine how many child tickets, c, and how many adult tickets, a, were sold.

• c+a=1470

• c+a=150

• 7.75+10.25

Multiple Choice

Which equation represents the total amount of money earned from ticket sales?

The Cinema sold a total of 150 tickets to a movie. Some of these were child tickets and the rest were adult tickets. A child ticket cost $7.75 and an adult ticket cost $10.25. If the cinema sold a total of $1470 worth of tickets, determine how many child tickets, c, and how many adult tickets, a, were sold.

• 7.75c + 10.25a= 1470

• 7.75a+ 10.25c=1470

• 7.75c + 10.25a= 150

Open Ended

Solve the system of equations.

You may use any method to solve this system of equations: graphing, substitution, elimination, or by calculator.

c + a = 150

7.75c + 10.25a = 1470