Problem 60
Question
Perform the indicated operations. When possible write down only the answer. $$\frac{x-y}{y-x} \cdot \frac{1}{2}$$
Step-by-Step Solution
Verified Answer
-\frac{1}{2}
1Step 1: Simplify the Fraction
Observe the fraction \(\frac{x-y}{y-x}\). Notice that \(y-x = -(x-y)\). Therefore, we can rewrite the fraction as \(\frac{x-y}{-(x-y)} = -1\).
2Step 2: Rewrite the Expression
Rewrite the entire expression using the simplified fraction result from Step 1: \(-1 \cdot \frac{1}{2}\).
3Step 3: Perform the Multiplication
Multiply \(-1\) by \(\frac{1}{2}\): \(-1 \cdot \frac{1}{2} = -\frac{1}{2}\).
Key Concepts
Simplifying FractionsMultiplication of FractionsNegative Sign Handling
Simplifying Fractions
Simplifying fractions is the process of reducing them to their simplest form. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD).
In the provided exercise, the fraction is \(\frac{x-y}{y-x}\).
To simplify it, notice that \((y-x)\) is the negative of \((x-y)\). Therefore, you can write \(\frac{x-y}{y-x} = \frac{x-y}{-(x-y)} = -1\).
So, the entire fraction simplifies to \(-1\). Understanding this concept helps in dealing with more complex expressions, making them easier to solve.
In the provided exercise, the fraction is \(\frac{x-y}{y-x}\).
To simplify it, notice that \((y-x)\) is the negative of \((x-y)\). Therefore, you can write \(\frac{x-y}{y-x} = \frac{x-y}{-(x-y)} = -1\).
So, the entire fraction simplifies to \(-1\). Understanding this concept helps in dealing with more complex expressions, making them easier to solve.
Multiplication of Fractions
Multiplying fractions involves multiplying the numerators together and the denominators together.
In the exercise, after simplifying, we get \(-1 \times \frac{1}{2}\).
To multiply, treat \(-1\) as \(\frac{-1}{1}\), so it becomes:
\( \frac{-1 \times 1}{1 \times 2} = -\frac{1}{2}\).
This straightforward method ensures you get the correct product of your fractions every time.
In the exercise, after simplifying, we get \(-1 \times \frac{1}{2}\).
To multiply, treat \(-1\) as \(\frac{-1}{1}\), so it becomes:
\( \frac{-1 \times 1}{1 \times 2} = -\frac{1}{2}\).
This straightforward method ensures you get the correct product of your fractions every time.
Negative Sign Handling
Handling negative signs correctly is critical in algebra. Misplacing a negative sign can lead to incorrect answers.
In our problem, we simplified \(\frac{x-y}{y-x}\) to \(-1\), noting that \((y-x)\) is the negative of \((x-y)\).
We then multiplied \(-1\) by \(\frac{1}{2}\) to get \(-\frac{1}{2}\).
The key point is to remember that a fraction with a negative statement in the denominator can be simplified by moving the negative sign out or to the numerator, ensuring the fraction stays valid.
In our problem, we simplified \(\frac{x-y}{y-x}\) to \(-1\), noting that \((y-x)\) is the negative of \((x-y)\).
We then multiplied \(-1\) by \(\frac{1}{2}\) to get \(-\frac{1}{2}\).
The key point is to remember that a fraction with a negative statement in the denominator can be simplified by moving the negative sign out or to the numerator, ensuring the fraction stays valid.
Other exercises in this chapter
Problem 59
Simplify. $$\left(x^{-1}+y^{-1}\right)^{-1}$$
View solution Problem 60
Solve each equation. $$\frac{-3}{2 x}=\frac{1}{-5}$$
View solution Problem 60
Simplify. $$\left(a^{-1}-b^{-1}\right)^{-2}$$
View solution Problem 61
Solve each equation. $$\frac{x}{x-2}-\frac{x+2}{x^{2}-2 x}=\frac{1}{x}$$
View solution