To solve for $P$ and $Q$, let's analyze the equation given in the image:

$(P+2, Q-1) = (2Q+1, P-2)$

This implies that we have two equations by matching the coordinates:

1. $P + 2 = 2Q + 1$
2. $Q - 1 = P - 2$

Now, let's solve these equations:

Step 1: Solve Equation (2) for $Q$ in terms of $P$

$Q - 1 = P - 2$ $Q = P - 1$

Step 2: Substitute $Q = P - 1$ into Equation (1)

$P + 2 = 2(P - 1) + 1$ $P + 2 = 2P - 2 + 1$ $P + 2 = 2P - 1$ $3 = P$

Step 3: Substitute $P = 3$ back into the equation for $Q$

$Q = P - 1 = 3 - 1 = 2$

Solution

The values of $P$ and $Q$ are: $P = 3, \quad Q = 2$

Would you like more details on any specific step or have further questions?

Here are some questions related to this solution:

1. How would the solution change if we had a different set of equations?
2. Can this problem be solved graphically?
3. Are there other methods to verify the solution?
4. What if we introduced another variable, how would it affect the solution?
5. How can we interpret this solution in a real-world context?

Tip: When dealing with systems of equations, substitution and elimination are powerful methods for finding solutions.