Math Problem Statement

(P + 2, Q - 1) = (2Q + 1, P - 2). Determine (P, Q).

Solution

To solve for PP and QQ, let's analyze the equation given in the image:

(P+2,Q1)=(2Q+1,P2)(P+2, Q-1) = (2Q+1, P-2)

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

  1. P+2=2Q+1P + 2 = 2Q + 1
  2. Q1=P2Q - 1 = P - 2

Now, let's solve these equations:

Step 1: Solve Equation (2) for QQ in terms of PP

Q1=P2Q - 1 = P - 2 Q=P1Q = P - 1

Step 2: Substitute Q=P1Q = P - 1 into Equation (1)

P+2=2(P1)+1P + 2 = 2(P - 1) + 1 P+2=2P2+1P + 2 = 2P - 2 + 1 P+2=2P1P + 2 = 2P - 1 3=P3 = P

Step 3: Substitute P=3P = 3 back into the equation for QQ

Q=P1=31=2Q = P - 1 = 3 - 1 = 2

Solution

The values of PP and QQ are: P=3,Q=2P = 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.

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Math Problem Analysis

Mathematical Concepts

Algebra
System of Equations
Substitution Method

Formulas

Equation of two coordinates

Theorems

Equality of coordinates

Suitable Grade Level

Grades 8-10