Problem 57 - Algebra for College Students

Question

Perform the indicated operations. When possible write down only the answer. $$\frac{x-y}{3} \cdot \frac{6}{y-x}$$

Step-by-Step Solution

Answer

-2

Step 1 - Identify the expression

The given expression is \(\frac{x-y}{3} \times \frac{6}{y-x}\).

Step 2 - Simplify the expression

Notice that \(y-x = -(x-y)\). So, replace \(y-x\) with \(-(x-y)\). The expression becomes \(\frac{x-y}{3} \times \frac{6}{-(x-y)}\).

Step 3 - Combine the fractions

Combine the fractions by multiplying the numerators and the denominators: \(\frac{(x-y) \times 6}{3 \times -(x-y)}\).

Step 4 - Cancel out common terms

Notice that \(x-y\) in the numerator and denominator cancel out, giving \(\frac{6}{3 \times -1}\).

Step 5 - Perform the final operation

Simplify \(\frac{6}{-3}\) to get \(-2\).

Key Concepts

simplifying expressions

Simplifying expressions is a fundamental step in solving algebraic problems. It involves making an expression as simple as possible by combining like terms, reducing fractions, and removing unnecessary parentheses.
In our example, we start by rewriting parts of the expression to make it easier to handle. Recognizing that \(y-x=- (x-y)\) helps us simplify the given fractions by adjusting the terms. This often involves reducing complex fractions or expressions into simpler parts.
As a quick tip, always look for patterns or equalities that can help simplify the terms. Keeping the expression as simple as possible means fewer steps in the final calculation!

fraction multiplication

Multiplying fractions might seem tricky, but it becomes straightforward once you know the steps. The process requires multiplying the numerators together and the denominators together.
Here’s how we do it in this exercise:

• First, identify the fractions from our expression: \(\frac{x-y}{3}\) and \(\frac{6}{y-x}\).
• Second, rewrite \(y-x\) as \(-(x-y)\).
• Third, substitute back into the expression to get: \(\frac{x-y}{3} \times \frac{6}{-(x-y)}\).
• Lastly, multiply the numerators and denominators: \(\frac{(x-y)\times 6}{3 \times -(x-y)}\).

Now we have one combined fraction that’s ready for any further simplification.

canceling terms

Canceling terms is an effective way to simplify fractions further. It involves reducing fractions by eliminating identical factors in the numerator and the denominator.
In our specific example:

• We start with the combined fraction \(\frac{(x-y)\times 6}{3 \times -(x-y)}\).
• Notice \(x-y\) appears in both the numerator and the denominator.
• By canceling out \(x-y\) from both, we simplify to \(\frac{6}{3 \times -1}\).
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This step significantly reduces our work and helps to ultimately find the final simplified result: \(\frac{6}{-3} = -2\). Always check for terms that can be canceled to avoid fully multiplying and then reducing again later. This helps save time and keeps calculations clean.