use Cramer's rule to solve the following
2x + 3y - z = 1
4x + y - 5z = 11
3x + 2y + 5z =21
To solve the system of equations using Cramer's rule, we need to find the determinants of the coefficient matrix and each of its associated matrices.
First, let's find the determinant of the coefficient matrix:
| 2 3 -1 |
| 4 1 -5 |
| 3 2 5 |
det(A) = 2(1*5 - (-5*2)) - 3(4*5 - (-5*3)) + (-1)(4*2 - 3*1)
det(A) = 2(5 + 10) - 3(20 + 15) - 1(8 - 3)
det(A) = 2(15) - 3(35) - 1(5)
det(A) = 30 - 105 - 5
det(A) = -80
Now, let's find the determinant of the matrix formed by replacing the x-coefficients with constants:
| 1 3 -1 |
| 11 1 -5 |
| 21 2 5 |
det(A1) = 1(1*5 - (-5*2)) - 3(11*5 - (-5*21)) + (-1)(11*2 - 21*1)
det(A1) = 1(5 + 10) - 3(55 + 105) - 1(22 - 21)
det(A1) = 1(15) - 3(160) - 1(1)
det(A1) = 15 - 480 - 1
det(A1) = -466
Next, let's find the determinant of the matrix formed by replacing the y-coefficients with constants:
| 2 1 -1 |
| 4 11 -5 |
| 3 21 5 |
det(A2) = 2(11*5 - (-5*21)) - 1(4*5 - (-5*3)) + (-1)(4*21 - 3*4)
det(A2) = 2(55 + 105) - 1(20 + 15) + (-1)(84 - 12)
det(A2) = 2(160) - 1(35) - 1(72)
det(A2) = 320 - 35 - 72
det(A2) = 213
Finally, let's find the determinant of the matrix formed by replacing the z-coefficients with constants:
| 2 3 1 |
| 4 1 11 |
| 3 2 21 |
det(A3) = 2(1*21 - 11*2) - 3(4*21 - 11*3) + 1(4*2 - 3*1)
det(A3) = 2(21 - 22) - 3(84 - 33) + 1(8 - 3)
det(A3) = 2(-1) - 3(51) + 1(5)
det(A3) = -2 - 153 + 5
det(A3) = -150
Next, let's find the values of x, y, and z using Cramer's rule:
x = det(A1) / det(A)
x = -466 / -80
x = 5.825
y = det(A2) / det(A)
y = 213 / -80
y = -2.6625
z = det(A3) / det(A)
z = -150 / -80
z = 1.875
Therefore, the solution to the system of equations is x = 5.825, y = -2.6625, z = 1.875.