Find the Value of P in the Equation
To find the value of P in the given equation, we need to solve for P using the given equation: p + 7 ÷ p - 6 = 1 ÷ 3.

Step 1: Simplify the Equation
First, let's simplify the given equation by dividing 7 by p and 1 by 3:
p + 7 ÷ p - 6 = 1 ÷ 3
p + 7/p - 6 = 1/3

Step 2: Cross Multiply
To eliminate the fractions, we can cross multiply:
3(p) + 3(7/p) - 3(6) = 1(p) - 1(6)
3p + 21/p - 18 = p - 6

Step 3: Combine Like Terms
Now, let's combine like terms on both sides of the equation:
3p + 21/p - 18 = p - 6
2p + 21/p = 12

Step 4: Solve for P
To solve for P, we can multiply both sides by p to get rid of the fraction:
2p^2 + 21 = 12p
2p^2 - 12p + 21 = 0

Step 5: Factor the Quadratic Equation
Factor the quadratic equation to solve for P:
(2p - 3)(p - 7) = 0

Step 6: Find the Values of P
Now, set each factor to zero and solve for P:
2p - 3 = 0 or p - 7 = 0
2p = 3 or p = 7
p = 3/2 or p = 7
Therefore, the values of P that satisfy the equation are p = 3/2 or p = 7.