use cramer's rule to solve for y

Question

x+4y-2z=3

x+3y+7z=1

2x+9y+z=8

Answer 1

Instead of contraction "det.",  I'll use Greek capital letter Δ (delta) for word determinant.
Cramer's rule is

          Δ_x
 x = ——— 
          Δ 

         Δ_y
y = ———
         Δ

         Δ_z
z = ———
         Δ

         1   4   -2   1   4   -2    
 Δ =   1   3    7   1   3    7
         2   9    1   2   9    1
It will be easy to find the determinant, if you will write it twice.
For first group, start multiplication from number on upper left corner and go down by diagonals.
For second group, start multiplication from the number on upper right corner and go down by diagonals.
Δ = 1 • 3 • 1 + 4 • 7 • 2 + (-2) • 1 • 9 – (-2) • 3 • 2 – 4 • 1 • 1 – 1 • 7 • 9 = - 14

To find Δ_x we have to replace first column (coefficients by "x") by numbers on the right sides of equations. We will take the same action for Δ_y and Δ_z

          3   4   - 2
Δ_x =   1   3     7
          8   9    1

          1   3   -2
Δ_y =   1   1    7
          2   8    1

          1   4    3 
Δ_z =    1   3    1 
          2   9    8 

Δ_x = 70 , Δ_y = - 28 , Δ_z = 0  

x = - 5 ,  y = 2 ,  z = 0

Answer 2

Let's first calculate the determinant of the matrix composed of coeffivients of the system:

 1   4   -2

  1   3   7

  2   9   2

This determinanat (A)equals

  Det      3    7               (-4) x  Det       1    7           + (-2) x Det     1     3

             9    2                                       2    2                                   2     9

It gives you:  Det A = - 15.

Now. To find x you have to replace all coefficients for x by the column composed of left- side

of the system:    3

                           1

                           8

It gives you new matrix (say, B)

                            3    4   -2

                            1     3   7

                            8     9   2

Calculate Det(B):   Det(B) = 75  and    x = Det(B)/Det(A) = -5. The same  way

y = Det (C)/Det (A), where matrix C is

     1      3      -3

     1      1       7

     2      8       2

Calculations show that Det (C) = - 30. Thus, y = -30/-15 = 2.

Likewise,  z = Det (D)/Det (A) where D is a matrix

  1       4        3

  1       3        1

  2       9        8

Det (D) = 0, therefore z = 0. Your solutions are:

                             x = -5       y = 2      z = 0

Plug these numbers into the system to make sure that the solutions are correct.