Question

Use Cramer's rule to find the value of z in the solution of the following system:

x−3y−z=−10
x−5y−5z=8
−2x−2y+2z=−68

Answer 1

To find the value of z using Cramer's rule, calculate the determinants of the coefficients matrix and the matrix obtained by replacing the coefficients of z with the constants, and then divide the determinant of z by the determinant of the coefficients.

Step-by-step explanation:

To use Cramer's rule to find the value of z in the given system of equations:

1. Identify the coefficients of x, y, and z, and the constants on the right side of each equation.
2. Calculate the determinant of the coefficients matrix, denoted as D.
3. Calculate the determinant of the matrix obtained by replacing the coefficients of z with the constants, denoted as Dz.
4. Find the value of z by dividing Dz by D.

For the given system of equations:

x - 3y - z = -10

x - 5y - 5z = 8

-2x - 2y + 2z = -68

The determinant D evaluates to -10, Dz evaluates to -5, and the value of z is -5/-10, which simplifies to 0.5.