Problem 19 - Algebra and Trigonometry

Question

Write an algebraic formula for the given quantity.. The sum \(S\) of an integer \(n\) and twice the integer

Step-by-Step Solution

Answer

The formula is \(S = 3n\).

Step 1: Identify the Integer and Express Twice the Integer

Let the integer be denoted by \(n\). To express twice the integer, multiply \(n\) by 2. This gives us \(2n\).

Step 2: Write the Formula for the Sum

The problem asks for the sum of the integer \(n\) and twice the integer. This sum can be formulated as: \(S = n + 2n\).

Step 3: Simplify the Formula

Combine like terms in the expression for \(S\). The terms \(n\) and \(2n\) are like terms and can be added together: \(S = 1n + 2n = 3n\).

Key Concepts

Integer Operations

In algebra, integer operations are the foundational building blocks that help us manipulate numbers within algebraic expressions. An integer is a whole number without a fractional or decimal component, such as 0, 1, 2, or -3. Operations with integers include addition, subtraction, multiplication, and division. Each of these operations follows specific rules, especially when negative integers are involved.

When working with the given problem, the operation performed is multiplication, where the integer variable is multiplied by 2 to achieve twice the integer; hence, we write it as \(2n\). Knowing how to effectively perform integer operations is crucial in solving algebraic expressions. Properly combining integers through these operations allows us to maintain the integrity of mathematical formulas.

Algebraic Formulas

Algebraic formulas are expressions used to represent quantities in a generalized form using variables and constants. Variables like \(n\) are symbols that represent unknown or variable numbers. In the exercise, we learned how an algebraic formula is constructed to represent the sum of an integer and twice that integer.

The formation of the formula begins by identifying each part of the expression. Here, the sum \(S\) is given by combining two quantities — the integer \(n\) and its double \(2n\). The algebraic formula is thus written as \(S = n + 2n\). Such formulas are valuable in mathematics as they can be applied to any integer value by substituting \(n\) with a specific number to find \(S\). Understanding algebraic formulas enables us to solve a wide range of mathematical problems by generalizing patterns and behaviors of numbers.

Like Terms Simplification

Simplifying expressions by combining like terms is an integral process in algebra. Like terms are those that have the same variable raised to the same power. In our exercise, the terms \(n\) and \(2n\) both contain the variable \(n\). Therefore, they are considered like terms and can be combined through addition.

The expression \(S = n + 2n\) is simplified by adding the coefficients of the like terms. Here, the coefficient of \(n\) is 1, and the coefficient of \(2n\) is 2. Adding these coefficients gives us \(3n\), resulting in the simplified form of the expression: \(S = 3n\).

Simplifying expressions makes them easier to work with and solve, as it reduces complexity and clarifies potential solutions. Emphasizing the process of combining like terms ensures efficiency when dealing with algebraic expressions in any mathematic scenario.