Problem 58 - College Algebra with Corequisite Support

Question

For the following exercises, use this scenario: The cost of renting a car is $$\$ 45 /$$ wk plus $$\$ 0.25 / \mathrm{mi}$$ traveled during that week. An equation to represent the cost would be \(y=45+.25 x,\) where \(x\) is the number of miles traveled. If your cost were $$\$ 63.75,$$ how many miles were you charged for traveling?

Step-by-Step Solution

Answer

75 miles were traveled.

Step 1: Understand the equation

The equation given is \( y = 45 + 0.25x \), where \( y \) represents the total cost, and \( x \) is the number of miles traveled. The goal is to find \( x \) when \( y = 63.75 \).

Step 2: Substitute the cost into the equation

Substitute \( y = 63.75 \) into the equation: \( 63.75 = 45 + 0.25x \). This will help you solve for \( x \), the number of miles.

Step 3: Isolate the variable x

Subtract 45 from both sides of the equation to isolate the term with \( x \):\[ 63.75 - 45 = 0.25x \].

Step 4: Simplify the equation

Perform the subtraction to simplify the left side of the equation: \( 18.75 = 0.25x \).

Step 5: Solve for x

Divide both sides of the equation by \( 0.25 \) to solve for \( x \): \[ x = \frac{18.75}{0.25} \].

Step 6: Calculate the value of x

Perform the division to find \( x \):\[ x = 75 \].

Step 7: Interpret the result

The result indicates that 75 miles were traveled to incur a cost of $63.75.

Key Concepts

Understanding the Cost Function in Linear Equations

In the context of our exercise, the cost function provides a direct way to calculate expenses related to renting a car. The equation given is \( y = 45 + 0.25x \) which is a linear expression. Here, \( y \) stands for the total cost you will pay after renting the car for a week, and \( x \) represents the miles traveled during that week.

A cost function can often be broken into two main parts: the fixed cost and the variable cost. In this case, \\(45 is the fixed cost, which does not change regardless of miles driven. The variable cost is \\)0.25 for every mile, which means the more you drive, the higher the total cost \( y \) will be.

Using such equations, you can easily predict or calculate the cost incurred from traveling a particular distance, which is crucial for budgeting and financial planning.

Solving Equations: Finding the Unknown

Solving equations is a fundamental skill in mathematics that involves finding the value of unknown variables. In our example, the unknown is the number of miles traveled \( x \). Solving this involves several straightforward steps that manipulate the equation to isolate \( x \).

Let's break this down:

• First, substitute the known value of \( y \) into the equation. Here, it is given as \\(63.75.
• Next, isolate the term containing \( x \) by subtracting the fixed cost \\)45 from both sides of the equation.
• Finally, you solve for \( x \) by dividing the remaining value by \$0.25.

By following these steps, the unknown value \( x \) can be determined accurately, allowing us to solve real-world problems effectively.

Insights into Algebraic Expressions

Algebraic expressions like \( y = 45 + 0.25x \) comprise constants, coefficients, and variables linked by operations. Let's explore these components further:

- Constants are fixed values that do not change. In our equation, \\(45 is the constant.- Coefficients are numerical or constant quantities placed before and multiplying a variable, which here is \\)0.25 associated with the mileage \( x \).- Variables represent the unknown values we are trying to find, labeled as \( x \) in this particular exercise.
Understanding these components is central to manipulating and solving various algebraic equations that rely heavily on these similar structures.

Recognizing Variables in Equations

Variables are symbols that stand for unknown values within equations, most commonly represented by letters such as \( x \) or \( y \). They are critical in forming equations and modeling real-life situations.

In our cost equation example, \( x \) represents the number of miles traveled, while \( y \) indicates the total cost. These variables allow flexibility, enabling the equation to adapt to different values, delivering precise calculations for varying scenarios.

Mastering variables in equations involves understanding their interaction with constants and coefficients, enabling one to translate mathematical expressions into practical solutions that help predict or assess various events or transactions.