Problem 79 - Algebra for College Students

Question

Solve each problem by using a system of equations. A cash drawer contains only five- and ten-dollar bills. There are 12 more five-dollar bills than ten-dollar bills. If the drawer contains \(\$ 330\), find the number of each kind of bill.

Step-by-Step Solution

Answer

There are 30 five-dollar bills and 18 ten-dollar bills.

Step 1: Define Variables

Let's define the variables we will use in our equations. Let \( x \) represent the number of five-dollar bills, and \( y \) represent the number of ten-dollar bills.

Step 2: Formulate Equations

Since there are 12 more five-dollar bills than ten-dollar bills, we can write the equation: \[ x = y + 12 \]Additionally, the total value of the bills in the cash drawer is $330. So, the value equation will be:\[ 5x + 10y = 330 \]

Step 3: Substitute and Solve for y

Substitute the expression for \( x \) from the first equation into the second equation:\[ 5(y + 12) + 10y = 330 \]Expanding gives:\[ 5y + 60 + 10y = 330 \]Combine like terms:\[ 15y + 60 = 330 \]Subtract 60 from both sides:\[ 15y = 270 \]Divide by 15:\[ y = 18 \]

Step 4: Solve for x

Now that we have \( y = 18 \), substitute back into the equation \( x = y + 12 \):\[ x = 18 + 12 \]\[ x = 30 \]

Step 5: Verify the Solution

Check that the total value of the bills matches the given total. With 30 five-dollar bills and 18 ten-dollar bills, calculate the total:\[ 5 \times 30 + 10 \times 18 = 150 + 180 = 330 \]The conditions are satisfied, so the solution is verified.

Key Concepts

Linear Equations

Linear equations are fundamental in algebra and involve expressions where each variable is raised to the power of one. This simplicity makes them one of the easiest types of equations to handle. In a linear equation, you can expect the graph to be a straight line, hence the name.
Linear equations can take many forms, but the most familiar is the slope-intercept form: \[ y = mx + b \], where \( m \) represents the slope and \( b \) the y-intercept.
When solving problems involving a linear system of equations, we often encounter scenarios that require us to find intersecting points of two lines, representing two conditions in the problem.
In our exercise, we see linear equations used to model the number and value of bills in a cash drawer, making use of conditions described in word problems. Understanding these details is crucial in setting up our equations correctly.

Substitution Method

The substitution method is a technique used to solve systems of equations, particularly effective when dealing with linear equations. It involves solving one of the equations for one variable and then substituting that expression into the other equation.
This method is practical because it reduces a system of equations to a single equation with one unknown, simplifying the problem-solving process.

• Solve one of the equations for one variable.
• Substitute this expression in place of the variable in the other equation.
• Solve the resulting single-variable equation.
• Use the solved value to find the other variable from the rewritten expression.

In the given exercise, substitution helps streamline calculations by transforming the system into a sequence of simpler operations. Step 3 of the solution is where the substitution primarily occurs, making it a plotting point in our approach to solving the problem.

Problem Solving

Problem solving in math not only involves finding a direct answer but developing a strategy to navigate through various types of information.
To solve a problem using systems of equations, you need to:

• Understand the problem and what is being asked.
• Translate the problem scenario into mathematical language — equations.
• Solve these equations using an appropriate method (e.g., substitution, elimination).
• Verify the solution to ensure it satisfies all parts of the original problem.

The original exercise involves understanding how the number of five-dollar bills relates to the ten-dollar bills and their combined value.
Accurate interpretation of these relationships is key for translating them into equations effectively, which then allows us to utilize algebraic methods, like substitution, to reach the solution.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. They play a crucial role in forming equations and are essential for solving equations in algebraic methods.
In any problem involving algebra, understanding how to manipulate these expressions is vital. Solving the given exercise required setting up expressions for the number of five-dollar and ten-dollar bills and their total value — key parts of the system of equations.

• Identify quantities and their relationships described in the problem.
• Form equations using these relationships.
• Manipulate the equations, using properties of equalities to simplify and solve.

Mastering the use of algebraic expressions enables you to break down complex problems into smaller, more manageable parts, much like solving a puzzle by finding each piece's exact fit in the bigger picture.