91Ó°ÊÓ

First, let's write the given system of linear equations in its matrix form AX = B, where A is the coefficient matrix, X is the column matrix of variables (x, y, z), and B is the column matrix of constants. \(A = \begin{bmatrix} 2 & 3 & 1 \\ 3 & -5 & 2 \\ -1 & -2 & 3 \end{bmatrix}\), \(X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}\), and \(B = \begin{bmatrix} 1 \\ 8 \\ 3 \end{bmatrix}\). Now, let's find the determinant of the coefficient matrix A, |A|: |A| = \( 2(-5*3-(-2)*2)+3(3*3-(-2)*2)+1((-1)*(-5)-3*3) \) |A| = \( 2(-15+4) + 3(9+4) + (-3+9) = 2(-11) + 3(13) + 6 = -22 + 39 + 6 \) |A| = \( 23 \) Since |A| ≠ 0, the system is consistent and has a unique solution.

Now, let's use Cramer's rule to find the values of x, y, and z. We will substitute the column B in the matrix A for each variable and calculate the determinant of the resulted matrices: 1) For x, replace the first column of A with B: \(A_x = \begin{bmatrix} 1 & 3 & 1 \\ 8 & -5 & 2 \\ 3 & -2 & 3 \end{bmatrix}\) |A_x| = \( 1((-5)*3-(-2)*2) + 3(8*3-2*3) + 1(8*(-2)-2*(-5)) \) |A_x| = \( 1(-15+4) + 3(24-6) + 1(-16+10) = -11 + 3(18) - 6 \) |A_x| = \( -11 + 54 - 6 = 37 \) 2) For y, replace the second column of A with B: \(A_y = \begin{bmatrix} 2 & 1 & 1 \\ 3 & 8 & 2 \\ -1 & 3 & 3 \end{bmatrix}\) |A_y| = \( 2(8*3-2*3) - 1(3*3-(-1)*2) + (-1)(3*(-1)-8*(-1)) \) |A_y| = \( 2(24-6) + (-1)(9+2) + (-1) (3+8) = 2(18) - 11 + (-11) \) |A_y| = \( 36 - 22 = 14 \) 3) For z, replace the third column of A with B: \(A_z = \begin{bmatrix} 2 & 3 & 1 \\ 3 & -5 & 8 \\ -1 & -2 & 3 \end{bmatrix}\) |A_z| = \( 2((-5)*3-(-2)*8) + 3(3*(-2)-8*(-1)) + 6 \) |A_z| = \( 2(-15+16) + 3(-6+8) + 6 = 2(1) + 3(2) \) |A_z| = \( 2 + 6 = 8 \)

Now that we have the determinants |A_x|, |A_y|, and |A_z|, we can find the values of x, y, and z by dividing these determinants by |A|: x = |A_x| / |A| = \( 37 / 23 \) y = |A_y| / |A| = \( 14 / 23 \) z = |A_z| / |A| = \( 8 / 23 \)

To check our solution, let's substitute the values of x, y, and z back into the original system of linear equations: 1) \(3y + 2x = z + 1 \rightarrow 3(14/23) + 2(37/23) = (8/23) + 1 \) 2) \(3x + 2z = 8 - 5y \rightarrow 3(37/23) + 2(8/23) = 8 - 5(14/23) \) 3) \(3z - 1 = x - 2y \rightarrow 3(8/23) - 1 = (37/23) - 2(14/23) \) Upon substituting the values of x, y, and z into the original system of equations, we find that all of the equations are true. So, the unique solution to the system of linear equations is: x = \( 37 / 23 \) y = \( 14 / 23 \) z = \( 8 / 23 \)