Problem 131 - Intermediate Algebra

Question

In the following exercises, translate to an algebraic expression and simplify if possible. the quotient of -6 and the sum of \(a\) and \(b\)

Step-by-Step Solution

Answer

\(+ -6 / (a + b) \)

Step 1: Identify the Components

First, identify the components of the expression based on the description. Here, the components are -6 (a negative number) and the sum of the variables a and b.

Step 2: Write the Sum

Write down the sum of the variables a and b, which is represented as \(+ a + b \).

Step 3: Write the Quotient

The quotient can be represented as a division of -6 by the sum of a and b. In algebraic form, it is written as \(+ -6 / (a + b) \).

Step 4: Simplify

Since the quotient is already simplified and no further reduction is possible, the final expression remains \(+ -6 / (a + b) \).

Key Concepts

Quotient

In algebra, a quotient represents the result of division. When we talk about the quotient in expressions, we're discussing how one quantity is divided by another. For instance, in the exercise, -6 is divided by the sum of two variables, a and b. This division is expressed using the quotient. This can visually be represented as:

\[-6 / (a + b)\]
To understand this, think of a quotient as the answer you get when you divide one number by another. When simplifying complex algebraic expressions, identifying the quotient correctly is crucial. A faulty identification can lead to errors in solving problems.

Variables

Variables in algebra are symbols that represent unknown values. Most often, variables are denoted by letters like a, b, x, and y. In our exercise, the sum of two variables, a and b, is crucial as it forms the divisor in our quotient.

It's important to understand that variables can take any value, which is why they are so powerful in algebra. They allow the expression to be general and can be applied to a wide range of problems. In real-world applications, variables help model scenarios where certain quantities are unknown or can change. Here are a few key points about variables:

• They represent unknowns in equations.
• They allow expressions to be more generalized.
• They can be manipulated algebraically to solve for unknowns.

Simplification

Simplification in algebra involves making an expression easier to work with without changing its value. It often includes combining like terms, reducing fractions, and performing basic arithmetic operations. In the exercise, the expression is:

\[-6 / (a + b)\]
Here, you start with the components—identifying the quotient and the sum of the variables. After writing down the initial expression, you check if further simplification is possible. For this particular exercise, no further simplification is needed beyond identifying and writing the quotient form.

Here are typical steps for simplification:

• Identify like terms and combine them.
• Reduce fractions to their simplest form.
• Factorize if possible to simplify the expression.

Simplifying correctly can make solving equations more manageable and help in understanding the underlying relationships within the problem.