The above equations have a unique solution if (a) p=2 ; q=3 (b) p=2 ; q ≠3 (c) p ≠2 ; q=3 (d)

step 1 So in the given question we have a system of equations that are given as 2x plus 2x plus p y plu

The above equations have a unique solution if (a) p=2 ;...

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Key Concepts

System of Linear Equations

A system of linear equations consists of multiple equations involving the same set of variables. The focus is on solving these equations simultaneously to find the values that satisfy all equations at once. Analyzing such systems helps determine whether there exists a unique solution, no solution, or infinitely many solutions, depending mainly on the structure and coefficients of the equations.

Determinant and Nonsingularity

For a system of linear equations represented in the matrix form, the determinant of the coefficient matrix plays a crucial role in determining whether a unique solution exists. A nonzero determinant means the matrix is invertible (nonsingular), which guarantees that the system has a unique solution. Conversely, a zero determinant indicates that the matrix is singular, leading to either no solution or infinitely many solutions.

Parameter Constraints in Linear Systems

When coefficients in a system of linear equations depend on parameters, these parameters directly influence the determinant of the coefficient matrix. Setting constraints on these parameters ensures that the conditions for a nonzero determinant are met, thereby guaranteeing a unique solution. This analysis is essential in understanding how variations in parameters affect the system's behavior and solution set.

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