Problem 53 - Algebra for College Students

Question

Perform the indicated operations. When possible write down only the answer. $$(a-b) \div(b-a)$$

Step-by-Step Solution

Answer

\(-1\)

Step 1: Recognize the Operation

Identify the given expression \((a-b) \div (b-a)\)

Step 2: Simplify the Expression

Notice that \(b - a\) can be written as \(-(a - b)\). So, substitute \((b-a)\) with \(-(a-b)\): \((a-b) \div -(a-b)\)

Step 3: Perform the Division

Divide \((a-b)\) by \(-(a-b)\): \((a-b) \div -(a-b) = \frac{a-b}{-(a-b)} = -1\)

Key Concepts

Operations with Variables

When working with algebraic expressions, you will often encounter variables, such as 'a' and 'b'. Variables represent unknown values and are critical components of algebra.
Operations with variables include addition, subtraction, multiplication, and division. Here are some key points to remember:

• Addition and subtraction: Combine like terms, which are terms with the same variable raised to the same power.
• Multiplication: Apply the distributive property, i.e., multiply each term in one polynomial by each term in the other.
• Division: Simplify the expression by canceling common factors when possible.

In our exercise, we are asked to perform a division operation on the given expression \(a - b\) \div \(b - a\).

Simplifying Expressions

Simplifying expressions is an essential skill in algebra. The goal is to rewrite the expression in its most reduced form. Let's focus on a few techniques to simplify expressions:

• Combine like terms: Add or subtract terms with the same variable and exponent.
• Factorization: Break down the expression into the product of its factors.
• Use identities: Recognize patterns like difference of squares, perfect squares, etc.

In the context of our exercise, we saw that \(b - a\) can be rewritten as \(\-(a - b)\). This simplification helps in making the problem more manageable. By substituting \(b - a\) with \(\-(a - b)\), the expression becomes straightforward to work with.

Division of Expressions

Division of algebraic expressions involves rewriting the ratio of two expressions to a simpler form. Here are some steps to keep in mind:

• Rewrite the division problem as a multiplication problem using the reciprocal of the divisor.
• Simplify the resulting expression by canceling out common factors when possible.

In our example, we examined \(a - b\) \div \(b - a\). By recognizing that \(b - a\) is the negative version of \(a - b\), we converted it to a multiplication problem: \(a - b\) \div \(\-(a - b)\). This makes the division straightforward and easy to solve.

Negative Reciprocals

Understanding negative reciprocals is crucial to the solution of the given exercise. A negative reciprocal occur when you take the negative of the reciprocal of a number or expression.
For instance, the reciprocal of \(a - b\) is \(\frac{1}{a - b}\), so the negative reciprocal would be \(\frac{-1}{a - b}\). In our exercise, we realized that \(b - a\) is the negative of \(a - b\), making \(b - a\) equal to \(\-(a - b)\). By using this property, we simplified \(a - b\) \div \(\-(a - b)\) which results in \frac{a-b}\-(a-b) = -1\. This negative reciprocal allowed us to quickly find the solution.