Finding Ordered Pairs (p,q) for Unique Solutions in a Linear System

Math Problem Statement

Number of ordered pairs (p,q) which system of equation has unique solution is

To determine the number of ordered pairs $(p, q)$ such that the system of equations has a unique solution, we need to analyze the system:

1) \quad px + qy &= 1, \\ 2) \quad qx + py &= 1. \end{aligned}$$ For a system of linear equations to have a unique solution, the determinant of the coefficient matrix must be non-zero. The coefficient matrix for this system is: $$\begin{pmatrix} p & q \\ q & p \end{pmatrix}.$$ The determinant of this matrix is: $$\text{Determinant} = \begin{vmatrix} p & q \\ q & p \end{vmatrix} = p \times p - q \times q = p^2 - q^2.$$ For the system to have a unique solution, the determinant must not be zero: $$p^2 - q^2 \neq 0.$$ This simplifies to: $$(p - q)(p + q) \neq 0.$$ Therefore, both $$p - q \neq 0$$ and $$p + q \neq 0$$, meaning: 1. $$p \neq q$$, 2. $$p \neq -q$$. Thus, $$p$$ and $$q$$ must be distinct and non-opposite numbers. If $$p$$ and $$q$$ can take any real values, then there are infinitely many pairs of $$(p, q)$$ that satisfy these conditions. Would you like more details on how to further analyze or extend this? Here are 5 related questions: 1. What happens if $$p = q$$ in the system of equations? 2. How would the system behave if $$p = -q$$? 3. Can you find a specific solution for any given values of $$p$$ and $$q$$? 4. How do we classify the system if the determinant equals zero? 5. How would this system change if the constant terms were different? **Tip**: The determinant of a 2x2 matrix provides a quick way to determine whether a system of linear equations has a unique solution, no solution, or infinitely many solutions.

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Linear Algebra
Determinants
Systems of Equations

Determinant of a 2x2 matrix: det(A) = ad - bc
p^2 - q^2

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