- Identify the denominators as \(1\), \(y^2-1\), and \(y+1\).
- Factor the expression \(y^2-1\) into \((y-1)(y+1)\).
- Determine that the Least Common Multiple (LCM) of the denominators is \(y^2-1\).
- Conclude that multiplying by \(y^2-1\) clears all denominators, making it the correct choice: \(\boxed{y^2 - 1}\).
Explanation
- Analyzing the Equation and Denominators
We are given the rational equation:
\[y + \frac{y^2 - 5}{y^2 - 1} = \frac{y^2 + y + 2}{y + 1}\]
Our objective is to determine the best expression to multiply both sides by in order to eliminate all denominators efficiently. The denominators present are \(1\) (associated with the term \(y\)), \(y^2 - 1\), and \(y + 1\). To simplify this equation, we look for the Least Common Multiple (LCM) of these denominators.
- Factoring and Determining the LCM
First, we factor the quadratic denominator \(y^2 - 1\). Using the difference of squares identity, we have:
\[y^2 - 1 = (y - 1)(y + 1)\]
The denominators are therefore \(1\), \((y - 1)(y + 1)\), and \((y + 1)\). The LCM of these expressions must contain all the unique factors that appear in the denominators, each raised to the highest power with which it appears. The factors are \((y - 1)\) and \((y + 1)\). Thus, the LCM is \((y - 1)(y + 1)\), which is equal to \(y^2 - 1\).
- Selecting the Correct Multiplier
We can now evaluate the given options: \(y\), \(y^2 - 1\), \(y + 1\), and \(y^2 + y + 2\). Multiplying both sides of the equation by \(y^2 - 1\) will effectively clear all denominators, as:
\[(y^2 - 1) \cdot \left( y + \frac{y^2 - 5}{y^2 - 1} \right) = (y^2 - 1) \cdot \left( \frac{y^2 + y + 2}{y + 1} \right)\]
\[y(y^2 - 1) + (y^2 - 5) = (y - 1)(y^2 + y + 2)\]
This confirms that \(y^2 - 1\) is the correct multiplier.
Examples
Identifying the correct common denominator is a fundamental skill in solving rational equations. In real-life scenarios, this process is akin to finding a common base for comparing different rates or proportions. For example, if you are comparing interest rates over different compounding periods or mixing ingredients with different concentration levels, you normalize these values using a common denominator to find the total or average, making it easier to solve for the unknown quantity effectively.