Janet wants to solve the equation [tex]y+\frac{y^2-5}{y^2-1}=\frac{y^2+y+2}{y+1}[/tex]. What should she multiply both sides of the equation by?
A. [tex]y[/tex]
B. [tex]y^2-1[/tex]
C. [tex]y+1[/tex]
D. [tex]y^2+y+2[/tex]
Answer (2)
Factor the denominator y 2 − 1 as ( y − 1 ) ( y + 1 ) .
Identify the least common multiple (LCM) of the denominators y 2 − 1 and y + 1 , which is y 2 − 1 .
Multiply both sides of the equation by the LCM to eliminate the denominators.
The expression Janet should multiply both sides of the equation by is y 2 − 1 .
Explanation
Analyze the equation First, let's analyze the given equation: y + y 2 − 1 y 2 − 5 = y + 1 y 2 + y + 2 We need to determine what expression Janet should multiply both sides of the equation by to eliminate the denominators.
Identify the denominators The denominators in the equation are y 2 − 1 and y + 1 . To eliminate these denominators, we need to find the least common multiple (LCM) of these expressions.
Factor the denominators We can factor y 2 − 1 as ( y + 1 ) ( y − 1 ) . Therefore, the denominators are ( y + 1 ) ( y − 1 ) and ( y + 1 ) .
Find the LCM The LCM of ( y + 1 ) ( y − 1 ) and ( y + 1 ) is ( y + 1 ) ( y − 1 ) , which is equal to y 2 − 1 .
Conclusion Therefore, Janet should multiply both sides of the equation by y 2 − 1 to eliminate the denominators.
Examples
When solving equations involving fractions, it's crucial to eliminate the denominators to simplify the equation. This problem demonstrates how to identify the least common multiple (LCM) of the denominators, which is the expression you need to multiply both sides of the equation by to clear the fractions. For instance, if you're distributing ingredients in a recipe and need to ensure each portion has the correct ratios, understanding how to find the LCM helps you scale the recipe accurately without changing the proportions.
Janet should multiply both sides of the equation by y 2 − 1 to eliminate the denominators. This is because y 2 − 1 is the least common multiple of the denominators present in the equation. The correct answer is option B, y 2 − 1 .
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