Need help answering the following using Cramer's Rule:

Question

2x+y-4z=-4

4x-3y+8z=-8

-2x+7y-12z=24

Answer 1

Dear Mr/Ms; this is my solution:

D = Matrix of 2    1     -4       you get this from the equation at the beginning system (2x + y -4z = -4;.....)      

 

                    4    -3     8

                    -2    7    -12

Thus, we have: matrix

D = 2* ( -3  8)      -  1* (4   8)      + -4* ( 4    -3)

           (  7   -12)          (-2  -12)         -2     7

 

D = 2*( (-12*-3) - 7*8) - 1* (-12*4 - -2*8) -4*(7*4 - -2*-3)

D = -96

Accordingly, we have Dx= ( -4  1   -4         this comes out when you apply the result of beginning to replace x position

                                        -8   -3   8

                                        24   7   -12)

Dx = -4* (-3  8)       - 1* (-8   8)      -4* (-8   -3

               7   -12)            24   -12            24   7)

Dx = -4*(-12*-3 -7*8) - 1*(-8*-12-24*8) -4*(-8 *7-24*-3)

Dx = 112

then, we establish matrix for Dy: (2   -4     -4

                                                 4    -8     8

                                                 -2   24   -12)

Dy= 2* (-8    8  )   - -4*( 4   8)   -4* (4   -8)

            24    -12)           -2  -12)      -2  24)

Dy = 2* (-12 *-8 - 24*8)  - - 4*(-12*4 - -2*8) - 4*(24*4 - (-2)*(-8))

Dy = -640

Accordingly, a matrix for Dz =   (2   1   -4

                                               4   -3   -8

                                               -2   7   24)

Dz = 2* ( -3   -8    -  1*( 4   -8  +  -4*(4    -3

               7   24)           -2   24)          -2   7)

Dz =  2*(-3*24  - 7*(-8))  - 1*(24*4)-(-8)*(-2)) -4*(7*4)-(-3)*(-2))

Dz = -200

Thus, x = Dx/D = 112/-96

        y = Dy/D = -640/-96

        z  =  Dz/D= -200/-96

Thanks for reading and you could reach me if you have more question. Thank you

 

 

Answer 2

"Use Cramer's Rule" means use determinants.

 

the system determinant will be D =

2  1  -4  

4 -3   8  

-2 7  -12 

and you can find its value in a variety of ways, including using the MATRIX menu of a 'graphing calculator.'

 

That is your job.  Next, replace the first column with the right-side constants of the three simultaneous equations of the determinant D to get D_x =

-4 1 -4 
-8 -3 8 
24 7 -12 

Then x = D_x/D.

 

Now, go back to the matrix that corresponded to D, and this time replace the y column by those right-side constants of the three simultaneous equations. Then calculate D_y and thus y = D_y/D

 

At this point you know x and y, so you can calculate z by substitution into any convenient equation.

 

DON'T FORGET TO CHECK YOUR ANSWERS.   BTW, if D = 0, there is no single (x,y,z) solution.