what is the solution to the equation $\frac{y}{y-4}-\frac{4}{y+4}=\frac{32}{y^2-16}$?

$y=-4$ and $y=4$
$y=0$
all real numbers
no solution

Question

what is the solution to the equation $\frac{y}{y-4}-\frac{4}{y+4}=\frac{32}{y^2-16}$?

$y=-4$ and $y=4$
$y=0$
all real numbers
no solution

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Least Common Denominator", "Solving Rational Equations", "Extraneous Solutions" ], "new_concepts": [], "current_concepts": [ "Least Common Denominator", "Solving Rational Equations", "Extraneous Solutions" ] } ## Identify the domain restrictions Using the *Extraneous Solutions* knowledge point $$ \begin{aligned} &y - 4 eq 0 \implies y eq 4 \ &y + 4 eq 0 \implies y eq -4 \ &y^2 - 16 eq 0 \implies y eq \pm 4 \end{aligned} $$ ## Find the least common denominator Using the *Least Common Denominator* knowledge point $$ \begin{aligned} &y^2 - 16 = (y - 4)(y + 4) \ &\text{LCD} = (y - 4)(y + 4) \end{aligned} $$ ## Clear the denominators and solve Using the *Solving Rational Equations* knowledge point $$ \begin{aligned} &y(y + 4) - 4(y - 4) = 32 \ &y^2 + 4y - 4y + 16 = 32 \ &y^2 + 16 = 32 \ &y^2 = 16 \ &y = \pm 4 \end{aligned} $$ ## Check for extraneous solutions Using the *Extraneous Solutions* knowledge point $$ \begin{aligned} &\text{Proposed solutions: } y = 4, \quad y = -4 \ &\text{Since } y eq \pm 4 \text{ due to domain restrictions, both values are extraneous.} \ &\text{Therefore, there is no solution.} \end{aligned} $$

$y = -4$ and $y = 4$

$y = 0$

all real numbers

no solution

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Solving Rational Equations" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the domain restrictions

Find the least common denominator

Clear the denominators and solve

Check for extraneous solutions

Answer: