Problem 131 - Algebra and Trigonometry

Question

Exercises \(131-133\) will help you prepare for the material covered in the next section. Jane's salary exceeds Jim's by \(\$ 150\) per week. If \(x\) represents Jim's weekly salary, write an algebraic expression that models Jane's weekly salary.

Step-by-Step Solution

Answer

The algebraic expression that represents Jane's weekly salary in relation to Jim's is \(x + 150\).

Step 1: Identify Variables

The problem involves two main variables: Jane's salary and Jim's salary. Jim's salary is represented by the variable \(x\). Let's represent Jane's salary as \(J\). It's stated that Jane's salary exceeds Jim's by $150 each week, treating these words 'exceeds by' as a mathematical operation it translates to addition.

Step 2: Formulate Algebraic Expression

We can substitute \(x\) for Jim's salary in the sentence 'Jane's salary exceeds Jim's by $150 per week' to form the algebraic expression. Thus, Jane's salary is given as \(J = x + 150\).

Step 3: Express the Algebraic Expression

Finally, express the algebraic expression that represents Jane's salary in relation to Jim's. So, when Jim's salary is expressed as \(x\), Jane's salary can be expressed as \(x + 150\).

Key Concepts

Variables in Algebra

When it comes to algebra, one of the fundamental concepts is the use of variables. Variables are symbols, usually letters, that stand in for unknown values. They can represent numbers in equations and expressions, serving as placeholders until the specific values they represent are determined.

In our exercise, we have identified two variables related to salaries: Jim's weekly salary, which is represented by the variable x, and Jane's weekly salary. These variables are pivotal because they allow us to describe mathematical relationships and solve problems even when we don't know the actual numbers.

It's key to choose variables that make sense contextually and make the algebraic statements easier to understand. For instance, using x to represent Jim's salary is fairly standard. Variables don't have to be a mystery; they're just a crucial part of the language of algebra.

Formulating Expressions

The ability to transform real-world scenarios into algebraic language is what formulating expressions is all about. This skill involves using variables, numbers, and mathematical operations to create an algebraic expression that models a given situation.

In our exercise, the given information needs to be translated into an expression that encapsulates Jane's salary in terms of Jim's. We interpret the phrase 'exceeds by' as a cue to use addition. Hence, by adding 150 to Jim's salary x, we represent Jane's salary algebraically. This process is fundamental as it bridges the gap between a written description and an algebraic equation which can be manipulated mathematically.

Why Is This Important?

Creating accurate algebraic expressions is a foundational ability in algebra. It enables students to tackle a variety of problems, from simple arithmetic to complex equations. Approaching real-life problems with algebra begins with this very step of formulating expressions that represent the essence of the problem.

Algebraic Representation

Once we have an established expression, we're engaging in what's called algebraic representation. This concept takes the numerically represented relationships between variables and encapsulates them into a formal algebraic expression or equation.

In the context of our problem, the relationship between Jane's and Jim's salaries is expressed as J = x + 150, where J represents Jane's salary and x represents Jim's. This algebraic representation is powerful—it can be used to calculate Jane's salary for any given value of Jim's salary, or be manipulated to find Jim's salary if Jane's is known.

Algebraic representation isn't just writing an equation; it's about understanding the linked components within a situation and using algebra as a tool to model and analyze those connections. This step is beyond formulating expressions—it is about solidifying the relationship in a precise mathematical form that can be universally understood and used to find solutions.