Here are the essential concepts you must grasp in order to answer the question correctly.
Cramer's Rule
Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, provided the determinant of the coefficient matrix is non-zero. It utilizes determinants to express the solution of each variable as a ratio of two determinants: the determinant of the coefficient matrix and the determinant of a modified matrix where one column is replaced by the constants from the equations.
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Determinants
A determinant is a scalar value that can be computed from the elements of a square matrix and provides important information about the matrix, such as whether it is invertible. For a 2x2 matrix, the determinant is calculated as ad - bc for a matrix of the form [[a, b], [c, d]]. If the determinant is zero, it indicates that the system of equations may have either no solution or infinitely many solutions.
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Alternative Methods for Solving Systems
When the determinant of the coefficient matrix is zero (D = 0), Cramer's Rule cannot be applied, and alternative methods must be used to find the solution set. These methods include substitution, elimination, or graphical representation. These techniques help determine whether the system has no solution (inconsistent) or infinitely many solutions (dependent), allowing for a comprehensive understanding of the system's behavior.
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