Use Cramer's rule to solve for u without solving x,y and z?

To use Cramer's rule to solve for u without solving for x, y, and z, we can set up the following system of equations:

2x + 3y + z = 10
x - 2y + 3z = 5
3x + y - 2z = 15

We can rewrite this system in matrix form as:

| 2 3 1 | | x | | 10 |
| 1 -2 3 | * | y | = | 5 |
| 3 1 -2 | | z | | 15 |

Now, we can use Cramer's rule to solve for u:

1. Calculate the determinant of the coefficient matrix:

| 2 3 1 |
| 1 -2 3 |
| 3 1 -2 |

det(A) = 2(-2)(-2) + 3(3)(3) + 1(1)(1) - 1(-2)(3) - 3(1)(2) - 2(3)(1)
det(A) = 4 + 27 + 1 + 6 - 3 - 6
det(A) = 29

2. Replace the first column of the coefficient matrix with the constants vector:

| 10 3 1 |
| 5 -2 3 |
| 15 1 -2 |

3. Calculate the determinant of this modified matrix:

det(Au) = 10(-2)(-2) + 3(3)(15) + 1(5)(1) - 1(-2)(15) - 3(1)(10) - 2(3)(5)
det(Au) = 40 + 135 + 5 + 30 - 30 - 30
det(Au) = 180

4. Calculate u:

u = det(Au) / det(A)
u = 180 / 29
u ≈ 6.21

Therefore, u ≈ 6.21.

$$[\text{det}(A)]u = [\text{det}(A_u)]$$