Use Cramer’s Rule to solve the following system of linear equations. Find Dx, Dy, Dz, and determinant. Give the value of y.

Question

4x - 3y + z = 29

6x + 9y - 2z = 6

3x - 2y + 4z = 31

Answer 1

Cramer's Rule is pretty cool, but you need to be able to find the determinant of a 3x3 matrix. D is the determinant of the matrix of coefficients. Dx, Dy, and Dz are the determinants of the matrices with the ith column (i = x, y, or z) substituted by the solution vector column. Once you have these, then

x = Dx/D, y = Dy/D, and z = Dz/D

determinant of the coefficient matrix is |4 -3 1|

|6 9 -2| = 4*9*4 +(-3*-2*3) + 1*6*-2 -3*9*1-(-2*-2*4)-4*-3*6 =

| 3 -2 4| 179

determinant, Dy is |4 29 1|

|6 6 -2| = 4*6*4+29*-2*3+1*6*31-1*6*3-4*-2*31-4*29*6=

|3 31 4| -358

Therefore y = (-358/179) = -2

By the way the solution vector (x y z) = (5 -2 3)

I'll leave the rest to you.

Answer 2

Make a matrix using the coeffecients of the equations

D=[4,-3,1] Determinent = 179

[6,9,-2]

[3,-2,4]

Next, make a matrix replacing the first column with the numbers on the right side of each equation

[29,-3, 1] Dx = 895

[6, 9,-2]

[31,-2, 4]

Make a matrix replacing the second column with the numbers on the right side for Dy

[4,29,1] Dy= -358

[6,6,-2]

[3,31,4]

Last make a matrix for Dz and replace the third column with the numbers on the right side

[4,-3,29] Dz= 537

[6, 9, 6]

[3,-2,31]

Now that you have the determinants for each matrix, you already have most of the question answered. In order to find the value of y, simply divide Dy/D, which gives you y = -2.

Therefore, by Cramer's rule, y = -2

Hope that helps!