Problem 250 - Elementary Algebra

Question

In the following exercises, translate to a system of equations and solve. A cashier has 54 bills, all of which are \(\$ 10\) or \(\$ 20\) bills. The total value of the money is \(\$ 910\). How many of each type of bill does the cashier have?

Step-by-Step Solution

Answer

17 ten-dollar bills and 37 twenty-dollar bills.

Step 1: Define variables

Let \( x \) be the number of \( \$10 \) bills and \( y \) be the number of \( \$20 \) bills.

Step 2: Set up the first equation

The cashier has 54 bills in total, so the equation is:\[ x + y = 54 \]

Step 3: Set up the second equation

The total value of the money is \( \$910 \), so the equation for the total value is:\[ 10x + 20y = 910 \]

Step 4: Simplify the second equation

Divide the second equation by 10 to simplify it:\[ x + 2y = 91 \]

Step 5: Solve the system of equations

We now have the following system of equations:\[ x + y = 54 \]\[ x + 2y = 91 \]Subtract the first equation from the second equation:\[ (x + 2y) - (x + y) = 91 - 54 \]\[ y = 37 \]

Step 6: Find the value of the other variable

Substitute \( y = 37 \) into the first equation to find \( x \):\[ x + 37 = 54 \]\[ x = 17 \]

Step 7: State the solution

The cashier has 17 ten-dollar bills and 37 twenty-dollar bills.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that represent relationships where each term is either a constant or the product of a constant and a single variable. They are straightforward and can be written in the form \[ ax + by = c. \]

These equations graph as straight lines and are foundational in algebra. In the exercise, we translated the problem into two linear equations:
\[ x + y = 54 \]
\[ 10x + 20y = 910 \]

Here, the variables are x and y, representing the number of ten-dollar and twenty-dollar bills respectively.

Solving Systems

Solving systems of equations means finding values for the variables that satisfy all equations simultaneously. Various methods can be used, such as graphing, substitution, or elimination. In our problem, we used the elimination method:

• Set up two equations from the problem statement.
• Simplify one or both equations if needed.
• Eliminate one variable by adding or subtracting the equations.
• Solve for the remaining variable.
• Substitute back to find the other variable.

In our case, the system of equations was:
\[ x + y = 54 \]
\[ x + 2y = 91 \]
After simplifying and subtracting the first from the second equation, we solved for y first and then substituted back to find x.

Variable Definition

Defining variables is a key initial step in solving word problems with systems of equations. A variable represents an unknown value that we aim to solve for.

In our exercise:

• Let's define x as the number of ten-dollar bills.
• Let y be the number of twenty-dollar bills.

Clear variable definitions help translate word problems into mathematical equations
Accurate definition ensures that the system of equations accurately reflects the problem's condition, allowing us to solve it correctly.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a real-world situation. It includes defining variables, forming equations, and solving them to determine unknown values. In the given problem:

• The cashier's bills and their total value were translated into a system of linear equations.
• The count of bills gave us one equation: \[ x + y = 54 \]
• The total value of the bills gave us the other: \[ 10x + 20y = 910\text{, simplified to } x + 2y = 91 \]

By forming and solving these equations, we modeled the problem mathematically, allowing us to determine the number of ten-dollar and twenty-dollar bills.