Question

Use Cramer's rule to solve the following set of equations: 2x + z + y = 1, 4x + y = -2, 2y - 2x + z = 4.

Answer 1

To solve the given set of equations using Cramer's rule, calculate the determinants of the coefficient matrix and the matrices formed by replacing each column with the column of constants. Using these determinants, solve for x, y, and z.

Step-by-step explanation:

To solve the given set of equations using Cramer's rule, we need to find the determinants of the coefficient matrix and the matrices formed by replacing each column with the column of constants.

1. Calculate the determinant of the coefficient matrix (D): D = |2 1 1| |4 1 0| |-2 -2 1| = -10.
2. Calculate the determinant of the matrix formed by replacing the first column with the column of constants (Dx): Dx = |1 1 1| |−2 1 0| |4 -2 1| = -9.
3. Calculate the determinant of the matrix formed by replacing the second column with the column of constants (Dy): Dy = |2 1 1| |4 -2 0| |-2 4 1| = -10.
4. Calculate the determinant of the matrix formed by replacing the third column with the column of constants (Dz): Dz = |2 1 1| |4 1 -2| |-2 -2 4| = -10.
5. Now, solve for x, y, and z: x = Dx/D = -9/-10 = 0.9, y = Dy/D = -10/-10 = 1, and z = Dz/D = -10/-10 = 1.