Problem 93 - Introductory and Intermediate Algebra for College Students 4th

Question

Let \(x\) represent the number. Express each sentence as a single algebraic expression. Then simplify the expression. Multiply a number by \(5 .\) Add 8 to this product. Subtract this sum from the number.

Step-by-Step Solution

Answer

The single algebraic expression is \(-4x - 8\).

Step 1: Express the Sentence in Algebraic Form

Let's translate each part of the sentence to an algebraic expression. 'Multiply a number by 5' can be expressed as \(5x\). 'Add 8 to this product' translates to \(5x + 8\). 'Subtract this sum from the number' turns the entire expression into \(x - (5x + 8)\).

Step 2: Simplify the Algebraic Expression

To simplify the expression, first deal with the subtraction. You want to subtract each term inside the parenthesis from \(x\), this gives \(x - 5x -8\).

Step 3: Simplify Further the Algebraic Expression

Further simplification involves combining like terms. \(x - 5x\) equals to \(-4x\). So, the final simplified algebraic expression is \(-4x - 8\).

Key Concepts

Simplifying Expressions

When simplifying expressions, the main goal is to reduce them to their simplest form. This often involves removing unnecessary parentheses, eliminating redundant terms, and breaking down wide-ranging expressions into more manageable parts. It becomes much easier to understand and solve algebraic expressions when they are simplified.

In the solution, we started with an expression that included complex operations: \((x - (5x + 8))\). The first step in simplifying was to distribute the negative sign in front of the parentheses to each term inside it. This means changing the expression from \((x - (5x + 8))\) to \(x - 5x - 8\).

Next, we continue to simplify by putting together like terms and reducing the expression even further. Eventually, the expression becomes easier to interpret and solve.

Combining Like Terms

Combining like terms is a fundamental concept in algebra that involves simplifying expressions by merging terms that have identical variables and powers. This helps to streamline expressions and makes calculations much more straightforward.

In the given expression, after removing the parentheses, we have: \(x - 5x - 8\). Here, the terms \(x\) and \(-5x\) are like terms because they both contain the variable \(x\) raised to the power of one.

• \(x\) essentially counts as \(1x\).
• \(-5x\) counts as subtracting the five from \(x\).

When you combine these, \(x - 5x\) results in \(-4x\). This merge of similar terms leads you towards a simpler expression: \(-4x - 8\).

As this concept reveals, focusing on merging like terms first can significantly simplify any daunting algebraic task.

Algebraic Translation

Translating sentences into algebraic expressions is crucial in solving real-world mathematical problems. Often, mathematical problems are expressed in everyday language rather than direct equations, requiring translation for better understanding and manipulation.

In our original exercise, the sentence 'Multiply a number by 5' becomes \(5x\) using algebraic translation, where \(x\) stands for the unknown number. This step is vital because it allows you to convert verbal descriptions into mathematical expressions that can be manipulated using algebra.

• 'Add 8 to this product' is translated as \(5x + 8\).
• 'Subtract this sum from the number' is expressed as \(x - (5x + 8)\).

Understanding this process makes it much easier to handle algebraic questions presented in everyday language, enhancing your problem-solving skills.