Problem 71 - Intermediate Algebra

Question

Use each of the variables \(a, b, c,\) and \(d\) only once to write: a. a sum of two differences b. a difference of two sums

Step-by-Step Solution

Answer

(a-b) + (c-d) is a sum of differences; (a+b) - (c+d) is a difference of sums.

Step 1: Understand the Exercise

The goal is to use given variables, \(a, b, c,\) and \(d\), to form specific expressions: one expression must be a sum comprised of two difference expressions, and one must be a difference comprised of two sum expressions.

Step 2: Construct a Sum of Two Differences

To construct a sum of two differences, create two separate difference expressions. For example, \( (a-b) \) and \( (c-d) \). Then, add them together: \[ (a-b) + (c-d). \]

Step 3: Construct a Difference of Two Sums

To construct a difference of two sums, start by creating two separate sum expressions. For example, \( (a+b) \) and \( (c+d) \). Then, subtract one from the other: \[ (a+b) - (c+d). \]

Key Concepts

Understanding a Sum of Differences

In algebra, a sum of differences is an expression where you first create differences using your variables and then add those differences together. Let's break it down with an example.
To create a sum of differences, you start by forming two difference expressions. For instance, using the variables provided:

• First difference: \( (a-b) \)
• Second difference: \( (c-d) \)

You then add these two differences together to create the sum of differences, represented as: \[(a-b) + (c-d)\]This type of expression is useful in various algebraic problems where you're asked to simplify or rearrange equations. It helps in situations where you need to consider the impact of changing quantities and their combined effect.

Exploring a Difference of Sums

The idea of a difference of sums involves creating sum expressions and then finding the difference between them. This concept is the reverse of the sum of differences we just explored.
To form a difference of sums, you need two sums first. Using our variables:

• First sum: \( (a+b) \)
• Second sum: \( (c+d) \)

Now, subtract the second sum from the first to form a difference of sums:\[(a+b) - (c+d)\]This technique is useful in scenarios where comparing collective quantities is necessary. It helps simplify problems where you need to find how two groups or sets differ from each other in total.

Delving into Variable Expressions

Variable expressions are fundamental in algebra as they allow you to represent numbers or quantities that can change. Each variable stands in for an unknown value, enabling flexible equations.
In our previous examples, the variables \(a, b, c,\) and \(d\) functioned as placeholders for actual numbers. Constructing expressions like \((a-b)\) or \((a+b)\) provides a way to describe possible changes or operations without specifying exact numbers.
Creating variable expressions is essential for solving equations and inequalities. When faced with a real-world problem, a variable can represent anything from the number of apples in a basket to the distance a car travels. These expressions become tools for modeling and solving complex problems, and understanding them can open the door to mastering algebra.