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.
$$\left\{\begin{aligned}
x+4 y-3 z &=0 \\
3 x-y+3 z &=0 \\
x+y+6 z &=0
\end{aligned}\right.$$

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Key Concepts

Conditions for Applicability of Cramer’s Rule

Cramer’s Rule is only applicable when the coefficient matrix is square and its determinant is non-zero. If the determinant is zero, the system does not have a unique solution, and Cramer’s Rule cannot be used to solve the system.

Cramer’s Rule

Cramer’s Rule is a method for solving systems of linear equations using determinants. The rule states that each variable in the system can be found by replacing the corresponding column of the coefficient matrix with the constants from the right-hand side, and then dividing the determinant of this modified matrix by the determinant of the coefficient matrix, provided that the latter is non-zero.

Determinant of a Matrix

The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether the matrix is invertible. For a system of equations, a non-zero determinant of the coefficient matrix indicates that the system has a unique solution.

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. Such systems can have a unique solution, infinitely many solutions, or no solution at all, and they are fundamental in linear algebra and various fields of science and engineering.

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