Problem 132 - Intermediate Algebra

Question

In the following exercises, translate to an algebraic expression and simplify if possible. the product of -13 and the difference of \(c\) and \(d\)

Step-by-Step Solution

Answer

The simplified expression is \-13c + 13d\.

Step 1: Identify the Components

Identify the terms given in the problem: - The product of -13- The difference of \(c\) and \(d\)

Step 2: Write the Difference of c and d

Write the difference as an algebraic expression: \(c - d\)

Step 3: Form the Product

Form the product of -13 and the difference \(c - d\): \-13 \times (c - d)\

Step 4: Simplify the Expression

Distribute -13 to both \(c\) and \(d\): \-13 \times (c - d) = -13c + 13d\

Key Concepts

algebraic expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations. Suppose you're solving a word problem. Translating the description into an algebraic expression is the first key step. Let's break down how: Consider the phrase 'the product of -13 and the difference of \(c\) and \(d\)'. Here:

• 'the product of' indicates multiplication.
• 'the difference of \(c\) and \(d\)' shows a subtraction where \(c\) is subtracted by \(d\).

By understanding these keywords, you can create proper algebraic expressions. In our example, it translates to: \[-13 \times (c - d)\]

simplification

Simplification in algebra means reducing expressions to their simplest form. After translating the word problem into the expression - \(-13 \times (c - d)\), let's simplify it. Simplifying often involves combining like terms, factoring, or distributing values. For our expression, the simplification step requires the distribution method. Apply the distributive property to simplify:
\-13 \times (c - d) = -13c + 13d\ Using this property makes expressions easier to handle. Always ensure each term within the parentheses interacts with the term outside.

distribution

Distribution is a powerful algebraic tool. It involves multiplying a single term by each term within a group (parenthesis). It's often indicated by expressions like \(a(b + c)\), which transforms into \(ab + ac\). In our example, applying distribution:

• Multiply -13 by \(c\)
• Multiply -13 by \(d\)

You'll get: \-13 \times (c - d) = -13c + 13d\ This method helps in breaking down complex expressions, making them manageable. Consistently practicing distribution will enhance your algebraic skills significantly!