Solving a System of Equations Using Cramers Rule

Using matrices and Cramer's Rule, solve for the values of $x$, $y$, and $z$ given the system of equations: $3x + 3y + 5z = 1$, $5x + 9y + 17z = 0$, $3x + 9y + 5z = 0$.

Cramer's Rule offers a unique method of solving systems of linear equations using determinants. This technique is particularly useful when dealing with a system in which the number of equations equals the number of unknowns, and the determinant of the coefficient matrix is non-zero, thereby ensuring the system has a unique solution. To apply Cramer's Rule, it's essential first to construct the coefficient matrix and calculate its determinant. This determinant serves as the divisor for the system's solution, provided it’s not equal to zero.

Once the determinant of the coefficient matrix is confirmed to be non-zero, the next step involves constructing matrices for each variable in the system, replacing the respective column of the coefficient matrix with the constants on the right-hand side of the equations. The determinants of these new matrices are then calculated as they relate directly to the values of the variables. Solving for each variable involves dividing the determinant of these modified matrices by the determinant of the original coefficient matrix. This approach elegantly encapsulates solving a system by focusing on determinants and matrix manipulations.

This problem not only reinforces the procedural application of Cramer's Rule but also highlights its limitations and areas of efficiency, particularly in smaller systems of linear equations where it remains computationally feasible. Utilizing determinants in this manner provides an excellent bridge into more advanced topics in linear algebra, such as matrix inverses and diagonalization, by emphasizing the importance of matrix manipulation and properties.