Using Cramer's rule, how do you solve?

2x-y=4
3x+y-z=10
y+z=3

To solve this system of equations using Cramer's rule, follow these steps:

Step 1: Set up the matrix equation
Write the equations in matrix form as AX = B, where A is the matrix of coefficients, X is the column matrix of the variables, and B is the column matrix of constants.

2 -1 0 x 4
3 1 -1 y 10
0 1 1 z 3

Step 2: Find the determinant of the coefficient matrix, |A|
Calculate the determinant of the coefficient matrix, |A| = 2(1)(1) + 3(1)(0) + 0(-1)(1) - 0(1)(1) - 2(1)(0) - 3(-1)(1) = 2 + 0 + 0 - 0 - 2 - (-3) = 3

Step 3: Find the determinants of X1, X2, and X3
Replace the first column of the coefficient matrix with the constant matrix B and find the determinant of the resulting matrix to calculate X1. Repeat this process for X2 and X3.

|B|
|A2| = 4 -1 0 x 4 = 4(1)(1) + 3(1)(1) + 0(x)(-1) - 0(1)(1) - 2(1)(x) - 3(-1)(y) = 7x + 3y - 2
|A3| 10 1 -1 y 10

|A1|
|B| = 2 -1 0 4 = 2(1)(1) + 0(1)(-1) + 3(y)(0) - 0(1)(4) - 2(-1)(y) - 3(0)(0) = 2y + 2
|A3| 3 1 -1 10 10

|A1|
|A2| 2 -1 0 4 = 2(1)(1) + 0(1)(1) + 3(y)(-1) - 0(1)(4) - 2(1)(4) - 3(-1)(10) = -7y - 5
|B|

Step 4: Calculate the values of x, y, and z
The solutions for x, y, and z can be found by dividing the determinants of X1, X2, and X3 by the determinant of the coefficient matrix.

x = |A2| / |A| = (-7y - 5) / 3
y = |A1| / |A| = (2y + 2) / 3
z = |A3| / |A| = (7x + 3y - 2) / 3

So, the solutions to the system of equations are:
x = (-7y - 5) / 3
y = (2y + 2) / 3
z = (7x + 3y - 2) / 3