To solve the given problem, we first need to understand the operation defined by p * q. According to the definition, p * q = 2p + pq + q. We need to find the value of p that satisfies the equation (p * 2) - (p * 1) = 40.
Let's break down the operations step by step:
- First, find p * 2:
p * 2 = 2p + p(2) + 2
Simplifying this, we get: p * 2 = 2p + 2p + 2 = 4p + 2
- Next, find p * 1:
p * 1 = 2p + p(1) + 1
Simplifying this, we get: p * 1 = 2p + p + 1 = 3p + 1
Using these simplifications, we substitute into the equation: (p * 2) - (p * 1) = 40
Substitute the expressions we found:
(4p + 2) - (3p + 1) = 40
Now, simplify the equation:
4p + 2 - 3p - 1 = 40
This simplifies to: 4p - 3p + 2 - 1 = 40
Further simplifying gives: p + 1 = 40
Subtract 1 from both sides to solve for p:
p = 40 - 1
p = 39
Therefore, the value of p is 39.