Problem 68 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. the sum of 8 times a number and twice the number

Step-by-Step Solution

Answer

The simplified algebraic expression is \(-20x\).

Step 1: Translate the English Phrase into an Algebraic Expression

To do this, you need to express the mathematical operations mentioned as symbols. The 'product of \(-9\) and a number' can be written as \(-9x\). The phrase 'increased by' can be translated as \(+\), and 'the product of \(-11\) and the number' can be written as \(-11x\). This gives the algebraic expression \(-9x + -11x\).

Step 2: Simplify the Algebraic Expression

We've translated the English phrase into the algebraic expression \(-9x + -11x\). You can simplify this by adding the two terms together. Both terms are like terms, as they have the same variable to the same power (in this case, \(x\) to the power of \(1\)). This simplification results in the algebraic expression \(-20x\).

Key Concepts

Algebraic Expression

An algebraic expression represents a mathematical phrase that includes numbers, variables, and operation symbols. The beauty of algebra lies in its ability to encapsulate complex ideas in a concise and manageable format. For example, in the exercise, the phrase 'the product of \( -9 \) and a number' is captured by \( -9x \), with \( x \) representing the unknown number and \( -9 \) denoting the coefficient that you multiply by the variable. Translating phrases like 'increased by' or 'decreased by' involves recognizing them as addition or subtraction signs, thus making the language of algebra a universal one. This translation is not just a word-for-word substitution but requires a comprehension of the underlying mathematical relationships. In simplifying these expressions, we distill the essence of the problem into a soluble algebraic equation, ready for further manipulation and solution.

Simplifying Expressions

The process of simplifying an expression is a key skill in algebra, making complex problems more manageable. It involves combining like terms, reducing fractions, and applying the distributive property when necessary. In our exercise, the expression \( -9x + -11x \) seems complex at first, but we quickly see that both terms involve the same variable to the same power. When simplifying, think of it like combining similar items in a basket; you're grouping the like items together to see what you're left with in total. After combining the coefficients \( -9 \) and \( -11 \) for the \( x \) terms, we obtain \( -20x \), a far simpler expression that is equivalent to our original, yet much easier to work with. Simplification is the art of making the complex simple, revealing the heart of the equation.

Like Terms

Imagine you have a collection of fruits in your basket. Apples can be grouped with apples, and oranges with oranges, but it would make no sense to mix apples with oranges when counting them. The concept of like terms in algebra functions similarly. Terms are considered 'like' if they contain the same variable raised to the same power. In algebraic expressions, like terms can be added or subtracted from each other. In our exercise, the terms \( -9x \) and \( -11x \) are like terms because they both contain the variable \( x \) to the first power. Recognizing like terms allows you to combine and simplify expressions, much like organizing a fruit basket streamlines your ability to understand what's inside. It's through identifying and combining like terms that we can simplify expressions, making them cleaner and more efficient for further operations.