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.
$$\left\{\begin{array}{r}
x-y+2 z=0 \\
3 x+2 y=0 \\
-2 x+2 y-4 z=0
\end{array}\right.$$

Key Concepts

System of Linear Equations

A system of linear equations consists of multiple linear equations that share common variables. The goal is to find the values of these variables that satisfy every equation in the system simultaneously. Such systems can have a unique solution, infinitely many solutions, or no solution, depending on the relationships among the equations.

Cramer’s Rule

Cramer’s Rule is a method for solving a system of linear equations using determinants. It provides an explicit formula for each variable, where the solution for each variable is given by the ratio of two determinants: one formed by replacing the corresponding column of the coefficient matrix with the constants from the right-hand side, and the other being the determinant of the coefficient matrix itself. This rule is applicable only when the determinant of the coefficient matrix is nonzero, ensuring a unique solution.

Determinant of the Coefficient Matrix

The determinant of the coefficient matrix is a scalar value that encapsulates key properties of the matrix. In the context of solving systems using Cramer’s Rule, a nonzero determinant indicates that the system has a unique solution since the matrix is invertible. Conversely, if the determinant is zero, the matrix is singular, meaning that the system may either have infinitely many solutions or no solution at all, and therefore Cramer’s Rule cannot be applied.

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