Help - struggling with infinite

Question

-2x-3y+2=-6

x+y-2=5

7x+8y-6z=31

Answer 1

Assuming that the second equation is x+y-2z=5, solve for x to get         x = 5-y+2z.

 

Substitute for x in the other two equations to obtain:

 

    -2(5-y+2z)-3y+2z=6 and 7(5-y+2z)+8y-6z=31

 

Simplify to get the system:  -y -2z = 16

                                         y+8z = -4

 

Adding the equations, we have 6z = 12.  So, z = 2

 

Since z = 2 and y+8z = -4, we get y+16 = -4  So, y = -20

 

Finally, since z = 2, y = -20 and x = 5-y+2z, we get x = 5+20+4 = 29

 

Solution:  x = 29, y = -20, z = 2.

 

Answer 2

I suppose you want a solution to the system of 3 linear equations in 3 unknowns.  Are you familiar with Cramer's rule?

 

Find the determinant of the coefficient matrix.  Call it D.

 

Find the determinants D_x, D_y, and D_z of the 3 matrices :

 

For D_x replace the first column of the coefficient matrix with the column of constants on the right.

 

For D_y replace the second column of the coefficient matrix with the column of constants on the right.

 

For D_z replace the third column of the coefficient matrix with the column of constants on the right.

 

Then x = D_x/D, y = D_y/D, z = D_z/D

 

Note that if D = 0, there is no solution.