Final answer:
To solve the given system of equations using Cramer's rule, calculate the determinant of the coefficient matrix. Then calculate the determinants of the matrices obtained by replacing each column with the constants. Finally, solve for the unknown variables using the determinants.
Step-by-step explanation:
To solve the given system of equations using Cramer's rule, we first need to calculate the determinant of the coefficient matrix. The coefficient matrix is:
1232-140-11
The determinant of this matrix is:
|A| = (1*(-1*1 - 4*(-1))) - (2*(2*1 - 4*0)) + (3*(2*(-1) - (-1)*0))
|A| = (-1 + 0 + 6) = 5
Next, we calculate the determinants of the matrices obtained by replacing the column of the coefficient matrix with the constant terms of the equations.
For the x-determinant (Ax), we replace the first column of the coefficient matrix with the constants:
52311-143-11
The determinant of this matrix is:
|Ax| = (5*(-1*1 - 4*(-1))) - (2*(11*1 - 4*3)) + (3*(11*(-1) - (-1)*3)
|Ax| = (-1 + 14 + 36) = 49
Similarly, for the y-determinant (Ay), we replace the second column of the coefficient matrix with the constants:
1532114031
The determinant of this matrix is:
|Ay| = (1*(11*1 - 4*3)) - (5*(2*1 - 4*0)) + (3*(2*3 - 11*0))
|Ay| = (11 - 10 + 6) = 7
Finally, for the z-determinant (Az), we replace the third column of the coefficient matrix with the constants:
1252-1110-13
The determinant of this matrix is:
|Az| = (1*(-1*3 - 11*(-1))) - (2*(2*3 - 11*0)) + (5*(2*(-1) - (-1)*0))
|Az| = (3 + 22 + 10) = 35
Finally, we can solve for the unknown variables using Cramer's rule:
x = |Ax| / |A| = 49/5 = 9.8
y = |Ay| / |A| = 7/5 = 1.4
z = |Az| / |A| = 35/5 = 7