In Problems 15- 12 , solve each system of equations using...

In Problems 15- 12 , solve each system of equations using Cramer's Rule if it is applicable. If Cranter's Rule is nor applicable, say sa
$\left\{\begin{array}{l}5 x-y=13 \\ 2 x+3 y=12\end{array}\right.$

Key Concepts

Applicability Conditions

For Cramer’s Rule to be applicable, the determinant of the coefficient matrix must be nonzero, ensuring that the system of equations has a unique solution. This condition is crucial because a zero determinant indicates either no solution or an infinite number of solutions, which falls outside the scope of Cramer’s Rule.

Cramer’s Rule

A method used to solve a system of linear equations that utilizes determinants to find the values of the variables. It provides an explicit formula for each variable as a ratio of determinants, and it is particularly useful when the system has the same number of equations as unknowns and the determinant of the coefficient matrix is not zero.

System of Linear Equations

A set of equations with multiple variables in which the goal is to find the values of these variables that satisfy all the equations simultaneously. This concept is fundamental in algebra and is applicable in many practical problems where different conditions must be met concurrently.

Determinant

A numerical value calculated from the elements of a square matrix that encapsulates several properties of the matrix, including the invertibility of the matrix and the uniqueness of the solutions to the system. A nonzero determinant indicates that the system has a unique solution while a zero determinant implies either no solution or infinitely many solutions.

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