Problem 57 - Prealgebra

Question

MP3 Players Southwest Electronics tracked the number of MP3 players it sold each month for a year. The store manager found that when he raised the price of the MP3 players just slightly, sales went down. He used the equation \(60=-2 x+130\) to determine the price \(x\) he needs to charge if he wants to sell 60 MP3 players a month. Solve this equation.

Step-by-Step Solution

Answer

The store needs to charge $35 to sell 60 MP3 players.

Step 1: Understand the equation

The equation provided to us is \(60 = -2x + 130\). This equation relates the number of MP3 players sold, which is 60, to the price \(x\) in dollars. Our task is to find the value of \(x\) when 60 MP3 players are sold.

Step 2: Isolate the linear term

We want to bring the terms with \(x\) to one side. Start by subtracting 130 from both sides of the equation to isolate the term containing \(x\) on one side.\[60 - 130 = -2x\]This simplifies to:\[-70 = -2x\]

Step 3: Solve for x

To solve for \(x\), divide both sides of the equation by \(-2\) to get \(x\) by itself.\[x = \frac{-70}{-2}\]Simplify this to:\[x = 35\]

Key Concepts

Solving Equations

Solving an equation involves finding a value for the variable that makes the equation true. In our case, we are working with a linear equation, which means it's the simplest type of equation we can solve.
You always start by thoroughly understanding the given equation, which for us is: \[60 = -2x + 130\]This equation signifies a relationship between the number of MP3 players sold per month and the price at which each is sold.

• Identify known and unknown quantities: Here, 60 is the number of MP3 players, and our task is to find the unknown price \(x\).
• Recognize the equation structure: It is a basic linear equation where terms are either a constant or a constant times a variable.

Solving the equation thus means rearranging or transforming it into a form where the unknown is isolated, allowing us to easily determine its value. This straightforward process lays the foundation for solving more complex equations in mathematics.

Variables

In mathematics, variables play a crucial role in forming equations. A variable is a symbol, usually a letter like \(x\), that represents an unknown or changeable component. In our exercise, the variable \(x\) represents the price necessary to ensure the sale of 60 MP3 players. Here’s how variables function in equations:

• Variables allow general representation of numbers, letting equations become versatile tools.
• They serve as placeholders that we solve for, as seen in the equation \(-2x + 130\). This term involves the variable \(x\) and constants \(-2\) and \(130\).

Understanding variables aids in forming real-life scenarios into problems we can solve analytically. With variables, problem-solving becomes an organized investigation of finding these unknowns, usually through algebraic techniques. Thus, variables transform daily situations into analyzable equations, making them invaluable in both mathematics and practical applications.

Equation Simplification

Equation simplification is an essential step in solving equations. It involves transforming the equation into its simplest form, which makes solving for the variable more straightforward. For our equation:\[60 = -2x + 130\]Simplification involves rearranging and reducing terms to isolate the variable \(x\). Here are the steps we followed:

• First, subtract \(130\) from both sides to eliminate constant terms on the variable's side. This gives us: \[-70 = -2x\].
• Then, divide both sides by \(-2\). This operation simplifies the equation to: \[x = 35\].

Each simplification step is carefully performed to maintain equation balance, ensuring no property of equality is lost. Simplification makes complex equations manageable and allows for clear, accurate determination of variable values. Once simplified, solving becomes an exercise in straightforward arithmetic, reaching the desired solution effectively.