Problem 11 - Algebra for College Students

Question

In Canada, \(\$ 1\) coins are called "loonies" because they have a picture of a loon on the reverse, and \(\$ 2\) coins are called "toonies." When Marissa returned home to San Francisco from a trip to Vancouver, she found that she had acquired 37 of these coins, with a total value of 51 Canadian dollars. How many coins of each denomination did she have?

Step-by-Step Solution

Answer

Marissa had 23 loonies and 14 toonies.

Step 1: Define Variables

Let \( x \) be the number of loonies (\$1 coins). Let \( y \) be the number of toonies (\$2 coins).

Step 2: Set Up Equations

Using the information provided, set up two equations based on the number of coins and their total value. The first equation is based on the number of coins: \[ x + y = 37 \] The second equation is based on the total value: \[ 1x + 2y = 51 \]

Step 3: Solve the System of Equations

First, solve the first equation for \( x \): \[ x = 37 - y \] Substitute this expression into the second equation: \[ 1(37 - y) + 2y = 51 \] Simplify and solve for \( y \): \[ 37 - y + 2y = 51 \]\[ y = 51 - 37 \]\[ y = 14 \]

Step 4: Find the Value of x

Substitute \( y = 14 \) back into the first equation: \[ x + 14 = 37 \]\[ x = 37 - 14 \]\[ x = 23 \]

Step 5: Verify the Solution

Check that the solution satisfies both original equations. For the number of coins: \[ 23 + 14 = 37 \] For the total value: \[ 1(23) + 2(14) = 23 + 28 = 51 \] Both equations are satisfied.

Key Concepts

Defining Variables

The first step in solving any word problem involving a system of linear equations is to define your variables. This means you decide what each variable will represent. In this example, we need to find the number of loonies and toonies Marissa has. Let's use the letter \( x \) to represent the number of loonies ( \$1 coins) and \( y \) to represent the number of toonies ( \$2 coins).

Choosing meaningful variables helps make our equations clearer and easier to solve. Think of \( x \) and \( y \) as placeholders for the unknown values we need to find.

Setting Up Equations

Once we define the variables, the next step is to convert the word problem into mathematical equations. We do this by using the information given in the problem.

The problem tells us:

• Marissa has 37 coins in total
• The total value of these coins is 51 Canadian dollars

We can use these two pieces of information to set up our equations. For the total number of coins, the equation is: \[ x + y = 37 \]
For the total value, since loonies are worth \$1 each and toonies are worth \$2 each, the equation is: \[ 1x + 2y = 51 \]

Solving Systems of Equations

Now that we have our system of equations, \( x + y = 37 \) and \( 1x + 2y = 51 \), we need to solve them to find the values of \( x \) and \( y \). There are different methods to solve systems of equations, such as graphing, elimination, and substitution.

In this exercise, we'll use the substitution method. First, we solve the first equation for \( x \): \[ x = 37 - y \]
Then, we substitute this expression into the second equation.

Substitution Method

Substitution is a method where we solve one of the equations for one variable and then substitute that expression into the other equation.

From the first equation, we already have: \[ x = 37 - y \] Now, we substitute \( 37 - y \) for \( x \) in the second equation: \[ 1(37 - y) + 2y = 51 \]
Simplifying this equation, we get: \[ 37 - y + 2y = 51 \] \[ 37 + y = 51 \] \[ y = 51 - 37 \] \[ y = 14 \]
Now, we have discovered that \( y = 14 \), or that Marissa has 14 toonies.

Verification of Solutions

The final step is to verify our solutions to ensure they satisfy both original equations.

We found \( y = 14 \). Substitute \( y \) back into the first equation to find \( x \): \[ x + 14 = 37 \] \[ x = 37 - 14 \] \[ x = 23 \]
Marissa has 23 loonies and 14 toonies.

Let's check these values in both equations:

• For the total number of coins: \( 23 + 14 = 37 \)
• For the total value: \( 1(23) + 2(14) = 23 + 28 = 51 \)

Both checks confirm that our solution is correct. Marissa indeed has 23 loonies and 14 toonies.