To solve the given system of equations using Cramer's rule, we need to find the values of x, y, and z in the equations:

1/x + 1/y = 3/2 ...(1)
1/y + 1/z = 5/6 ...(2)
1/z + 1/x = 4/3 ...(3)

Step 1: Write the Augmented Matrix

We start by writing the augmented matrix for the given system of equations. The augmented matrix consists of the coefficients of the variables on the left side and the constants on the right side.

The augmented matrix for the given system is:

[ 1/x, 1/y, 0 | 3/2 ]
[ 0, 1/y, 1/z | 5/6 ]
[ 1/z, 0, 1/x | 4/3 ]

Step 2: Find the Determinant of the Coefficient Matrix

The determinant of the coefficient matrix is calculated by using the coefficients of the variables in the equations.

The coefficient matrix is:

[ 1/x, 1/y, 0 ]
[ 0, 1/y, 1/z ]
[ 1/z, 0, 1/x ]

The determinant of the coefficient matrix is given by:

D = (1/x)(1/y)(1/x) + (1/y)(0)(1/z) + (0)(1/z)(1/y) - (0)(1/y)(1/x) - (1/x)(1/z)(0) - (1/z)(0)(1/y)

Simplifying the above expression, we get:

D = 1/(x^2y^2) + 1/(yz) - 1/(x^2z^2) - 1/(xy)

Step 3: Find the Determinants of the Variable Matrices

To find the determinants of the variable matrices, we replace the corresponding column of the coefficient matrix with the constants matrix.

For x determinant, we replace the first column with the constants matrix:

Dx = (3/2)(1/y)(1/x) + (5/6)(0)(1/z) + (4/3)(1/z)(0) - (0)(1/y)(1/x) - (1/y)(1/z)(0) - (1/z)(0)(1/y)

Simplifying the above expression, we get:

Dx = 3/(2xy) - 4/(3xz)

Similarly, for y determinant, we replace the second column with the constants matrix:

Dy = (1/x)(1/y)(1/x) + (0)(5/6)(1/z) + (1/z)(1/y)(0) - (0)(1/y)(1/x) - (1/x)(0)(1/z) - (1/z)(1/y)(0)

Simplifying the above expression, we get:

Dy = 1/(x^2y^2) + 1/(xz) - 1/(z^2y^2)

For z determinant, we replace the third column with the constants matrix:

Dz = (1/x)(1/y)(3/2) + (0)(1/y)(5/6)