Solve the following equation using Cramer's rule
x+2y-z=9
2x-y+3z=-2
3x+2y+3z=9
Cramer's rule involves solving a system of linear equations by using determinants. The given system of equations is:
1. x + 2y - z = 9
2. 2x - y + 3z = -2
3. 3x + 2y + 3z = 9
Let's define the coefficients matrix A and the constants matrix B:
A = | 1 2 -1 |
| 2 -1 3 |
| 3 2 3 |
B = | 9 |
| -2 |
| 9 |
The determinant of matrix A (D) is -14. Now, we will calculate the determinants of matrices obtained by replacing each column of A with matrix B, one column at a time:
Dx = | 9 2 -1 |
| -2 -1 3 |
| 9 2 3 |
Dy = | 1 9 -1 |
| 2 -2 3 |
| 3 9 3 |
Dz = | 1 2 9 |
| 2 -1 -2 |
| 3 2 9 |
Now, solve for Dx, Dy, and Dz:
Dx = 14
Dy = -49
Dz = -35
Finally, the solutions are given by:
x = Dx / D = 14 / -14 = -1
y = Dy / D = -49 / -14 = 3.5
z = Dz / D = -35 / -14 = 2.5
So, the solutions to the system of equations are x = -1, y = 3.5, and z = 2.5 using Cramer's rule.
