In Problems 15-42, solve each system of equations using...

In Problems $15-42,$ solve each system of equations using Cramer's Rule if it is applicable. If Cramer's Rule is not applicable, say so.
$$\left\{\begin{array}{l}
5 x-y=13 \\
2 x+3 y=12
\end{array}\right.$$

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Key Concepts

System_of_Linear_Equations

A system of linear equations consists of two or more equations that share common variables and are assumed to be simultaneously true. The goal is to find the values of the variables that satisfy all the equations in the system. This concept is fundamental in algebra and forms the basis for modeling and solving problems in various scientific and engineering disciplines.

Cramers_Rule

Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, provided that the coefficient matrix of the system has a non-zero determinant. It involves calculating determinants of matrices formed by replacing one column of the coefficient matrix with the constant terms from the equations. This rule provides explicit formulas for the solution and highlights the relationship between linear equations and matrix determinants.

Determinant

The determinant is a scalar value derived from a square matrix which indicates whether the matrix is invertible. In the context of systems of equations, a non-zero determinant signals that the system has a unique solution by Cramer's Rule, while a zero determinant indicates that the system is either inconsistent or has infinitely many solutions. Understanding the determinant is crucial for applying methods like Cramer's Rule effectively.

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