Question

Use the following information. At the start of your second year as a veterinary technician, you receive a raise of \(\$750.\) You expect to receive the same raise every year. Your total yearly salary after your first raise is \(\$ 18,000\) per year. Write an equation that models your total salary \(s\) in terms of the number of years \(n\) since you started as a technician.

Short answer

The equation that models your total salary \(s\) in terms of the number of years \(n\) since you started as a technician is \(s = 17250 + 750(n-1)\).

Step-by-step solution

Step 1. Understand the problem

The problem mentions that for the second year, a raise of \$750 is given taking the total salary to \$18,000. It is also stated that this same raise is expected each year.

Step 2. Determine the initial salary

The initial (first-year) salary will be \$18,000 (the total second-year salary) subtracted by the annual increment of \$750, which gives the initial salary as \$17,250 per year.

Step 3. Formulate the equation.

The total salary \(s\) for any given year can be calculated by adding the initial salary (\$17,250) to the annual increment (\$750) multiplied by the given year \(n-1\) (where \(n\) is the year number since the employment started). In mathematical terms, this can be translated into the following linear equation: \(s = 17250 + 750(n-1)\).

Key concepts

Understanding Algebraic Expressions

Algebraic expressions are a foundational concept in algebra. They consist of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. For instance, in the problem about salary increases, we refer to the total yearly salary as an algebraic expression that changes based on the number of years worked.
An expression like \(s = 17250 + 750(n-1)\) includes constants (\(17250\) and \(750\)), a variable (\(n\)), and operations (multiplication and addition). It's crucial to understand this because the correct manipulation of these expressions enables us to model real-world scenarios and solve problems algebraically. When approaching algebraic expressions, always identify the constants, variables, and operations involved. This clear identification will lay the groundwork for constructing and deconstructing complex expressions, making algebra more approachable.

Writing Equations

When writing equations, the aim is to create a mathematical statement that represents a situation or problem. Equations consist of two expressions set equal to each other. In the exercise provided, the equation to represent the salary situation is written based on the given information. The steps taken to write the equation followed these guidelines:

• Identify the variable: In this case, \(n\) represents the number of years since starting the job.
• Understand how the variable affects the situation: The salary increases by a fixed amount each year after the first.
• Determine constants: The raise amount (\(750\)) and the initial salary (\(17250\)) are constants, as they do not change in the context of the problem.
• Combine these components into an equation: The initial salary plus the raise multiplied by (\(n-1\)) (since the raise starts from the second year).

Providing clarity at each step ensures that anyone can follow the logic behind the equation and apply this process to similar problems. The ability to write equations accurately is one of the most valuable skills in algebra, as it serves as the starting point for problem-solving.

Arithmetic Sequences

Arithmetic sequences are ordered lists of numbers where the difference between consecutive terms is constant. This problem highlights a real-life application of arithmetic sequences, where the salary increase can be thought of as a sequence with a common difference. Each year, the raise stays the same, adding \(750\) to the previous year's salary.
Here's a closer look at the arithmetic sequence in the context of this problem:

• The first term (\(a_1\)) is the initial salary without any raise, which is calculated to be \(17250\).
• The common difference (\(d\)) is the annual raise which, according to the problem, is \(750\).
• Any subsequent term can be found using the formula \(a_n = a_1 + (n-1)d\), where \(a_n\) is the salary in the \(nth\) year.

Understanding arithmetic sequences not only aids in solving problems involving regular increments or decrements but also builds a foundation for more advanced topics in mathematics such as series and calculus.