Problem 74 - College Algebra

Question

Use a system of linear equations with two variables and two equations to solve. CDs cost $$\$ 5.96$$ more than DVDs at All Bets Are Off Electronics. How much would 6 CDs and 2 DVDs cost if 5 CDs and 2 DVDs cost $$\$ 127.73 ?$$

Step-by-Step Solution

Answer

The total cost of 6 CDs and 2 DVDs is \( \$147.68 \).

Step 1: Define the Variables

Let \( x \) represent the cost of one CD, and \( y \) represent the cost of one DVD.

Step 2: Set Up the Equations

From the problem statement, we know that CDs cost \( \\(5.96 \) more than DVDs. Thus, the first equation is \( x = y + 5.96 \). We also know the cost of 5 CDs and 2 DVDs is \( \\)127.73 \), which creates the second equation: \( 5x + 2y = 127.73 \).

Step 3: Substitute to Solve for One Variable

Substitute the expression for \( x \) from the first equation into the second equation: \[ 5(y + 5.96) + 2y = 127.73 \].

Step 4: Simplify the Equation

Expand and simplify the equation: \[ 5y + 29.8 + 2y = 127.73 \]. Combine like terms to get: \[ 7y + 29.8 = 127.73 \].

Step 5: Solve for \( y \)

Subtract 29.8 from both sides: \[ 7y = 97.93 \]. Divide both sides by 7 to find \( y \): \[ y = 13.99 \].

Step 6: Solve for \( x \)

Use the first equation to find \( x \): \[ x = y + 5.96 = 13.99 + 5.96 = 19.95 \].

Step 7: Calculate the Cost of 6 CDs and 2 DVDs

Using the values of \( x \) and \( y \), calculate the total cost of 6 CDs and 2 DVDs: \[ 6x + 2y = 6(19.95) + 2(13.99) \].

Step 8: Simplify to Find the Total Cost

Calculate: \[ 6 imes 19.95 = 119.70 \] and \[ 2 imes 13.99 = 27.98 \]. Add these two amounts to get the total cost: \[ 119.70 + 27.98 = 147.68 \].

Key Concepts

Two Variables

In the context of this problem, we are dealing with two unknown quantities, namely the cost of one CD and the cost of one DVD. These are our two variables, which we can denote as \( x \) and \( y \). Assigning variables to these quantities is the first step in setting up a system of equations.

By defining these variables clearly, we give ourselves a way to represent the relationships between CDs and DVDs mathematically.

• \( x \): Represents the cost of one CD.
• \( y \): Represents the cost of one DVD.

By knowing what these variables stand for, it sets the foundation for crafting equations that express these relationships. Understanding these relationships is crucial for solving our system of equations.

Substitution Method

The substitution method is a way of solving a system of linear equations where you solve one equation for one variable and then substitute that expression into the other equation.

In our problem, we first express \( x \), the cost of a CD, in terms of \( y \), the cost of a DVD by using the given information: \( x = y + 5.96 \). This equation reflects the relationship that CDs cost $5.96 more than DVDs.

Next, we substitute this expression into the second equation that describes the total cost of 5 CDs and 2 DVDs: \( 5x + 2y = 127.73 \).

By replacing \( x \) with \( y + 5.96 \), we have:

• \( 5(y + 5.96) + 2y = 127.73 \)

This step reduces our system to a single variable, allowing us to solve for \( y \). This substitution makes solving the equations more straightforward.

Solving Linear Equations

Solving linear equations involves simplifying the equation and isolating the variable to find its value. Once we've substituted and simplified our equation to involve only one variable, \( y \), the next step is to solve for \( y \).

We simplify our substitution equation as follows:

• \( 5y + 29.8 + 2y = 127.73 \)
• Combine like terms to get: \( 7y + 29.8 = 127.73 \)

The goal is to isolate \( y \). First, subtract 29.8 from both sides:

• \( 7y = 97.93 \)

Next, divide both sides by 7 to solve for \( y \):

• \( y = 13.99 \)

Now that we have \( y \), the cost of one DVD, we can easily find \( x \) by substituting the value of \( y \) back into the equation \( x = y + 5.96 \). Hence, \( x = 19.95 \).

Cost Calculation

Cost calculation is an important step where we apply the values we've found to determine the total expense for the items in question. Here, we need to calculate the cost of 6 CDs and 2 DVDs using our previously found values of \( x \) and \( y \).

The formula used is \( 6x + 2y \). Insert the costs:

• \( 6 \, \times \,19.95 = 119.70 \)
• \( 2 \, \times \,13.99 = 27.98 \)

Adding these two costs together gives the total cost:

• \( 119.70 + 27.98 = 147.68 \)

This total, $147.68, represents the complete cost for purchasing 6 CDs and 2 DVDs at the store. Thus, understanding how to apply the solution of a system of equations to practical situations like this can help in various real-life scenarios.