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
$\left\{\begin{array}{r}3 x-5 y=3 \\ 15 x+5 y=21\end{array}\right.$

Key Concepts

Coefficient Matrix and Invertibility

The coefficient matrix of a system of linear equations contains all the coefficients of the variables in the system. A key property of this matrix is its invertibility, which is determined by its determinant. A nonzero determinant indicates that the matrix is invertible, meaning the system has a unique solution, and techniques like Cramer’s Rule can be applied effectively.

Determinant

The determinant is a scalar value computed from the elements of a square matrix and provides critical information about the matrix, such as whether it is invertible. In the context of solving systems of equations, if the determinant of the coefficient matrix is nonzero, it ensures the existence of a unique solution, which is a necessary condition for applying Cramer’s Rule.

System of Linear Equations

A system of linear equations is a set of equations where each is of the first degree, and the solution is a set of values for the variables that satisfies all equations simultaneously. This concept is at the heart of linear algebra and is used to model and solve problems in various fields such as physics, engineering, and economics.

Cramer’s Rule

Cramer’s Rule is a method used to solve a system of linear equations with as many equations as unknowns by employing determinants. It expresses each variable as the ratio of the determinant of a modified coefficient matrix (where one column is replaced by the constants from the equations) to the determinant of the original coefficient matrix. This rule is especially useful for small systems where the determinant can be computed easily.

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