Questions: Solve the rational equation. Express numbers as integers or simplified fractions. p/(p-5) + 5/6 = 2/(p-5) The solution set is .

$Solve the rational equation. Express numbers as integers or simplified fractions. p/(p-5) + 5/6 = 2/(p-5) The solution set is .$

Transcript text: Solve the rational equation. Express numbers as integers or simplified fractions. \[ \frac{p}{p-5}+\frac{5}{6}=\frac{2}{p-5} \] The solution set is $\square$ . Check

Solution

Solution Steps

To solve the given rational equation, we first identify a common denominator for all terms, which is $p-5$. We then multiply each term by this common denominator to eliminate the fractions. After simplifying, we solve the resulting linear equation for $p$. Finally, we check the solution to ensure it does not make any denominator zero.

Step 1: Set Up the Equation

We start with the rational equation: \[ \frac{p}{p-5} + \frac{5}{6} = \frac{2}{p-5} \] To eliminate the fractions, we identify the common denominator, which is $p-5$.

Step 2: Eliminate the Denominator

Multiplying each term by $p-5$ gives us: \[ p + \frac{5}{6}(p-5) = 2 \] This simplifies to: \[ p + \frac{5p - 25}{6} = 2 \]

Step 3: Solve for $p$

Next, we multiply through by 6 to eliminate the fraction: \[ 6p + 5p - 25 = 12 \] Combining like terms results in: \[ 11p - 25 = 12 \] Adding 25 to both sides yields: \[ 11p = 37 \] Dividing by 11 gives: \[ p = \frac{37}{11} \approx 3.3636 \]

Step 4: Check for Validity

We must ensure that our solution does not make the denominator zero. Since $p - 5 \neq 0$ when $p = \frac{37}{11}$, this solution is valid.