Problem 35 - Algebra 1: Concepts and Skills

Question

You and a friend have 30 dollars to spend at a health center. It costs 10 dollars an hour to use the racquetball court and 5 dollars an hour to use the tennis court. Which equation represents the number of hours you can spend on each court? Let \(x\) represent the number of hours on the racquetball court and \(y\) represent the number of hours on the tennis court. A. \(5 x+10 y=30\) B. \(10 x+5 y=30\) C. \(5 y=10 x-30\) D. \(y=5 x+6\)

Step-by-Step Solution

Answer

Option B: \(10x+5y=30\) is the correct equation. It signifies spending $10 per hour on the racquetball court and $5 per hour on the tennis court which matches the conditions given in the problem.

Step 1: Analyzing options

Let's interpret all options one by one.

Step 2: Option A

If we look at option A \(5x+10y=30\), it implies that $5 is spent per hour on the racquetball court and $10 is spent per hour on the tennis court. This contradicts the actual costs given in the question.

Step 3: Option B

Option B on the other hand \(10x+5y=30\), implies that $10 is spent per hour on the racquetball court and $5 is spent per hour on the tennis court. This is directly in line with the costs given in the problem.

Step 4: Option C

Option C: \(5y=10x-30\), if re-arranged results in \(5x+5y=30\), which distorts the cost ratio between the racquetball and tennis courts. This also contradicts the costs given in the question.

Step 5: Option D

Option D: \(y=5x+6\), indicates that the number of hours on the tennis court is based on the number of hours on the racquetball court, plus 6. This doesn't consider the actual cost per hour and the total available cash, therefore, it doesn't represent the situation correctly.

Key Concepts

Algebraic Equations

In the realm of mathematics, an algebraic equation can be perceived as a scale, where each side of the equation contains expressions that are balanced through the equality sign. This concept becomes vividly illustrated when managing financial constraints, such as budgeting for activities like racquetball and tennis court rentals.

By definition, an algebraic equation is composed of variables, numbers, and an equality sign. The variables signify unknown values we endeavor to ascertain. In our health center scenario, these unknowns are the hours spent on each court, represented by the variables x and y. Algebraic equations manifest in various forms, from simple linear equations to more complex polynomial equations. The exercise at hand deals particularly with a linear equation - a type of algebraic equation where each term is either a constant or the product of a constant and a single variable.

Linear Equations

Linear equations form the backbone of algebra and are the foundation of a system of equations. They are named for their characteristic property—producing a straight line when graphed on a coordinate plane. A linear equation with two variables, like the one in our textbook problem involving a health center budget, can be expressed in the form ax + by = c, where a, b, and c are constants, and x and y are variables representing quantities to be determined.

The beauty of linear equations lies in their simplicity and their applicability to real-world problems. Let's consider Option B from the exercise, 10x + 5y = 30. Here, 10 and 5 represent the hourly costs of renting the racquetball and tennis courts respectively, while x and y denote the hours you can spend at each. Understanding and constructing linear equations are essential skills in algebra, enabling us to solve for unknowns in a clear and structured manner.

Problem-Solving in Algebra

Algebraic problem-solving is akin to a detective's work, where logic, reasoning, and the process of elimination help uncover the truth. When grappling with algebraic problems, breaking down the information into manageable pieces is crucial. For example, in our exercise, discerning the correct equation required analyzing each option against the given information.

Approaching algebraic problems often involves the following steps:

• Understanding the problem and identifying the relevant information
• Choosing the correct algebraic method to address the problem
• Organizing information into equations that represent real-world relationships
• Solving the equations using algebraic rules and techniques
• Checking the solutions to ensure they meet the problem's requirements

Logical analysis, as exhibited in the step-by-step dissection of each option, averts common pitfalls and guides us towards the correct choice, exemplified by Option B being the proper representation of the scenario.