Question

solve the rational equation. express numbers as integers or simplified fractions.
\\(\frac{1}{3p} + \frac{1}{4} = \frac{13}{6p} - \frac{2}{3}\\)

the solution set is \\(\\{\\ \\}\\).

Explanation:

Step1: Find common denominator (12p)

Multiply each term by $12p$ to eliminate denominators:
$$12p \cdot \frac{1}{3p} + 12p \cdot \frac{1}{4} = 12p \cdot \frac{13}{6p} - 12p \cdot \frac{2}{3}$$

Step2: Simplify each term

$$4 + 3p = 26 - 8p$$

Step3: Isolate p terms on left

Add $8p$ to both sides:
$$4 + 11p = 26$$

Step4: Isolate constant terms on right

Subtract 4 from both sides:
$$11p = 22$$

Step5: Solve for p

Divide both sides by 11:
$$p = \frac{22}{11} = 2$$

Step6: Verify solution

Substitute $p=2$ into original equation:
$\frac{1}{3(2)}+\frac{1}{4}=\frac{1}{6}+\frac{1}{4}=\frac{2+3}{12}=\frac{5}{12}$; $\frac{13}{6(2)}-\frac{2}{3}=\frac{13}{12}-\frac{8}{12}=\frac{5}{12}$. Both sides are equal, so $p=2$ is valid.

Answer:

$\{2\}$