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Chapter 4: Problem 18

Last month, Jerry Papa purchased five DVDs and two CDs at Wall-to-Wall Sound for 65 dollars. This month he bought three DVDs and four CDs for 81 dollars. Find the price of each DVD, and find the price of each CD.

Short Answer

Expert verified
Each DVD costs 7 dollars, and each CD costs 15 dollars.

Step by step solution

01

Define Variables

First, let's define the variables for the problem. Let \( x \) represent the price of one DVD and \( y \) represent the price of one CD.
02

Set Up Equations

Using the information given, we can set up two equations based on Jerry Papa's purchases. For last month, when Jerry bought 5 DVDs and 2 CDs for 65 dollars, the equation is \( 5x + 2y = 65 \). For this month, when he bought 3 DVDs and 4 CDs for 81 dollars, the equation is \( 3x + 4y = 81 \).
03

Solve the System of Equations - Eliminate One Variable

To eliminate one of the variables, let's multiply the first equation by 2 and the second equation by 1, to align the coefficients of \( y \): \( 10x + 4y = 130 \) and \( 3x + 4y = 81 \). Subtract the second equation from the first: \( (10x + 4y) - (3x + 4y) = 130 - 81 \), leading to \( 7x = 49 \).
04

Solve for x

From the equation \( 7x = 49 \), solve for \( x \) by dividing both sides by 7: \( x = 7 \). This means each DVD costs 7 dollars.
05

Solve for y - Substitute Back

Now that we know \( x = 7 \), substitute it back into one of the original equations, for example, \( 5x + 2y = 65 \): \( 5(7) + 2y = 65 \). Simplify to \( 35 + 2y = 65 \). Then, solve for \( y \) as follows: \( 2y = 65 - 35 \), so \( 2y = 30 \). Finally, \( y = 15 \). Thus, each CD costs 15 dollars.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
A linear equation is a type of equation that creates a straight line when graphed. This makes them easier to visualize and solve. Linear equations have one or more variables, but these variables are always to the power of one. They follow the standard format of:
  • \( ax + by = c \)
where "\( a \)" and "\( b \)" are coefficients, and "\( c \)" is a constant term. In our problem, the two linear equations are:
  • \( 5x + 2y = 65 \)
  • \( 3x + 4y = 81 \)
These equations help us understand the relationship between the number of DVDs and CDs Jerry bought and their respective costs. By solving them, we can find out how much one unit of each costs. Linear equations are often used in real-world problems involving constant rates or linear relationships.
Problem Solving
Problem solving involves approaching a question methodically to find a solution. When dealing with linear equations, these are the steps to follow:
  • Define the variables: In our context, let's identify what needs to be found, which are the prices of DVDs and CDs.
  • Translate into equations: Create an equation from each statement in the problem. In Jerry's case, these equations are based on his purchases and spending.
  • Solve using a particular method: Systems of equations can be solved using substitution, elimination, or even graphical methods. Here, we used elimination.
  • Verify your answer: Double check back your solution by plugging it back into the original equations.
By systematically following these steps, problem solving becomes structured and manageable. It helps in transforming complex problems into simpler, solvable ones.
Variables and Expressions
In algebra, variables are symbols that represent unknown numbers or values. In Jerry's DVD and CD problem, the variables are \( x \) and \( y \), representing the price of a DVD and a CD respectively. These variables are connected through expressions and equations.Expressions are combinations of variables, numbers, and operations (like addition or multiplication). For instance, \( 5x + 2y \) from the problem is an algebraic expression, which shows how different quantities are related.These expressions form the basis of the equations we solve. By manipulating expressions through operations like addition, subtraction, multiplication, or division, we can uncover the values of the variables they contain. Understanding how to work with variables and expressions is key to solving not just this problem, but many mathematical challenges.

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Most popular questions from this chapter

Ann Marie Jones has been pricing Amtrak train fares for a group trip to New York. Three adults and four children must pay 159 dollars. Two adults and three children must pay 112 dollars. Find the price of an adult's ticket, and find the price of a child's ticket.

The sum of the digits of a three-digit number is \(15 .\) The tens-place digit is twice the hundreds-place digit, and the ones-place digit is 1 less than the hundreds-place digit. Find the three-digit number.

Without graphing, decide. See Examples 7 and \(8 .\) a. Are the graphs of the equations identical lines, parallel lines, or lines intersecting at a single point? b. How many solutions does the system have? $$ \left\\{\begin{array}{l} {4 x+y=24} \\ {x+2 y=2} \end{array}\right. $$

While it is said that trains opened up the American West to settlement, U.S. railroad miles have been on the decline for decades. On the other hand, the miles of roads in the U.S. highway system have been increasing. The function \(y=-1379.4 x+150,604\) represents the U.S. railroad miles, while the function \(y=478.4 x+157.838\) models the number of U.S. highway miles, where \(x\) is the number of years after \(1995 .\) For each function, \(x\) is the number of years after \(1995 .\) (Source: Association of American Railroads, Federal Highway Administration). a. Explain how the decrease in railroad miles can be verified by their given function while the increase in highway miles can be verified by their given function. b. Find the year in which it is estimated that the number of U.S. railroad miles and the number of U.S. highway miles were the same. (IMAGE CANNOT COPY)

The planning department of Abstract Office Supplies has been asked to determine whether the company should introduce a new computer desk next year. The department estimates that 6000 dollars of new manufacturing equipment will need to be purchased and that the cost of constructing each desk will be 200 dollars. The department also estimates that the revenue from each desk will be 450 dollars. a. Determine the revenue function \(R(x)\) from the sale of \(x\) desks. b. Determine the cost function \(C(x)\) for manufacturing \(x\) desks. c. Find the break-even point.
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