1Cademy - Solving $$\frac{y}{y+6} = \frac{72}{y^2-36} + 4$$

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Strategy to Solve Equations with Rational Expressions

Example

Solving $\frac{y}{y+6} = \frac{72}{y^2-36} + 4$

Solve the rational equation $\frac{y}{y+6} = \frac{72}{y^2-36} + 4$ by applying the five-step strategy for equations with rational expressions. This example demonstrates a case where one of the two algebraic solutions turns out to be extraneous and must be discarded.

Step 1 — Identify restricted values. Factor the quadratic denominator using the difference of squares pattern: $y^2 - 36 = (y-6)(y+6)$ . Setting each factor equal to zero gives $y = 6$ and $y = -6$ . Record $y \neq 6$ and $y \neq -6$ .

Step 2 — Find the LCD. The factored denominators are $(y+6)$ and $(y-6)(y+6)$ . Since the quadratic denominator already contains $(y+6)$ as a factor, the LCD is $(y-6)(y+6)$ .

Step 3 — Clear the fractions. Multiply every term on both sides by the LCD $(y-6)(y+6)$ and cancel matching denominator factors:

$(y-6)(y+6) \cdot \frac{y}{y+6} = (y-6)(y+6) \cdot \frac{72}{(y-6)(y+6)} + (y-6)(y+6) \cdot 4$

Simplify each term: on the left, $(y+6)$ cancels, leaving $y(y-6)$ . On the right, the first term's entire denominator cancels, leaving $72$ , and the second term becomes $4(y-6)(y+6) = 4(y^2 - 36)$ :

$y(y-6) = 72 + 4(y^2 - 36)$

Step 4 — Solve the resulting equation. Distribute on both sides: $y^2 - 6y = 72 + 4y^2 - 144$ . Simplify the right side: $y^2 - 6y = 4y^2 - 72$ . Move all terms to one side: $0 = 3y^2 + 6y - 72$ . Factor out the GCF of $3$ : $0 = 3(y^2 + 2y - 24)$ . Factor the trinomial: $0 = 3(y+6)(y-4)$ . Apply the Zero Product Property:

$y + 6 = 0 \quad \text{or} \quad y - 4 = 0$

$y = -6 \quad \text{or} \quad y = 4$

Step 5 — Check. The value $y = -6$ equals one of the restricted values recorded in Step 1, so it is an extraneous solution and must be discarded — substituting $y = -6$ would make the denominators $(y+6)$ and $(y^2 - 36)$ equal zero.

Check $y = 4$ in the original equation:

$\frac{4}{4+6} = \frac{72}{4^2-36} + 4$

$\frac{4}{10} = \frac{72}{-20} + 4$

$\frac{4}{10} = -\frac{36}{10} + \frac{40}{10}$

$\frac{4}{10} = \frac{4}{10}$ ✓

The solution is $y = 4$ . This example illustrates a rational equation in which factoring the quadratic denominator $y^2 - 36$ as $(y-6)(y+6)$ is the key first step. The resulting quadratic produces two candidate solutions, but one of them — $y = -6$ — coincides with a restricted value and is therefore extraneous. Unlike earlier examples where all solutions were valid, here the checking step eliminates one candidate, leaving a single valid solution.

Updated 2026-04-30

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