Problem 54 - Algebra 1

Question

In Exercises 54 and 55 , your Internet service provider charges \(\$ 4.95\) per month for the first 3 hours of service, plus \(\$ 2.50\) for each additional hour. Your total charges last month were \(\$ 21.83 .\) Let \(x\) represent the number of hours you used the service. Which equation models the situation? A. \(4.95+2.5 x-3=21.83\) B. \(4.95+2.5(x-3)=21.83\)

Step-by-Step Solution

Answer

The equation that correctly models the situation is \(4.95+2.5(x-3)=21.83\).

Step 1: Understand the options

Option A, \(4.95+2.5 x-3=21.83\), represents a situation where for every hour, a charge of $2.5 is added, and then 3 (either hours or a deduction amount without units) is subtracted. This equation doesn't make sense in the context of the problem because it's not clear why we'd be subtracting 3. Neither hours nor a fixed amount were mentioned in the problem. Option B, \(4.95+2.5(x-3)=21.83\), represents the ISP's charging method more accurately: a flat charge for the first 3 hours, and then an additional charge for each hour beyond those initial three.

Step 2: Formulate whether the first 3 hours are included in the 'additional charges' or not.

The ISP's charging method includes a flat charge of $4.95 for the first 3 hours. Only after these 3 hours, a charge of $2.5 per hour is added. This is represented by the figure in the brackets, \(x-3\), in option B. This suggests that any hours beyond the first 3 hours are considered as 'additional'. Thus, the ISP's charging method, as mentioned in the problem, is correctly portrayed by option B.

Key Concepts

Algebra Word Problems

Understanding algebra word problems begins with the ability to translate a real-world scenario into mathematical expressions and equations. In the context of the exercise provided, the student must discern the charging method of an Internet service provider. The problem thus reflects one of the essential skills in algebra: interpreting verbal descriptions and identifying the correct algebraic representation.

The tactic to tackle such problems usually involves identifying the fixed and variable aspects of the scenario. The fixed cost in this instance is the base fee for the first three hours of service, while the variable cost is dependent on the number of additional hours used. Deciphering the correct model for the situation requires careful consideration of the language used in the description as well as logical reasoning to avoid common pitfalls, such as improperly setting up the equation or misinterpreting the charging structure.

For students struggling with word problems, it is crucial to practice the translation of words into numbers and operations regularly. This can be achieved by identifying keywords, determining constants and variables, and setting up equations step by step to reflect the situation accurately.

Systems of Equations

While the current exercise does not involve multiple equations, the concept of systems of equations is often closely related to algebra word problems. A system of equations consists of two or more equations with a different set of variables. The goal is to find the values of the variables that solve all the equations in the system simultaneously.

In more complex scenarios than the given exercise, you might be given multiple conditions that can be modeled by different equations, which when combined, form a system. For instance, if an additional piece of information related to another service with its own pricing structure were presented, you would then have two equations with a common variable to solve.

To achieve mastery of systems of equations, students should become familiar with methods such as graphing, substitution, and elimination. Each method has its own strengths and is suited to different kinds of systems. By practicing these techniques on a variety of problems, students can develop a better intuition for which method to use in a given situation.

Algebraic Expressions

Algebraic expressions are at the heart of any algebraic endeavor, including solving word problems. An expression is a combination of variables, numbers, and operations (like addition and multiplication) that represents a specific value or set of values. In the given exercise, the expression representing the additional charges beyond the first three hours is given as \(2.5(x-3)\).The parenthesis in this expression indicates that the subtraction should be performed before multiplication, which is in line with the arithmetic principle known as the order of operations. This order, often memorized using the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), guides the sequence in which the operations should be executed within an expression.It's vital for students to practice creating and simplifying expressions to become proficient with algebra. This includes operations like distributing (applying multiplication across a sum within parentheses), combining like terms (terms with the same variable raised to the same power), and factoring (rewriting an expression as a product of its factors). Each of these skills helps in solving algebraic expressions and understanding the algebraic structure underlying word problems.