Using determinants, solve the following equations simultaneously. 5x — 6y + 4z = 15 7x + 4y — 3z = 19 2x + y + 6z = 46

Answer Details

To solve the given system of equations using determinants, we need to form a matrix of coefficients of the variables and then calculate its determinant. Let's call this matrix A.

A =

|5 -6 4|
|7 4 -3|
|2 1 6|

We also need to form a matrix B by replacing the column of constants with the column of variables.

B =

|15 -6 4|
|19 4 -3|
|46 1 6|

Finally, we can solve for x, y, and z by using Cramer's rule, which says that x, y, and z are given by the ratios of the determinants of matrices formed by replacing the columns of A with the column of B, divided by the determinant of A.

x = |15 -6 4| / |5 -6 4| = 2

y = |5 15 4| / |5 -6 4| = 3

z = |-30 -30 90| / |5 -6 4| = 6

Therefore, the solution to the given system of equations is x = 2, y = 3, and z = 6.

In summary, we formed the matrix A of coefficients, the matrix B by replacing the column of constants with the column of variables, and used Cramer's rule to solve for x, y, and z by finding the ratios of the determinants.