how to solve a 4 variable equations

Question

4w+x+2y-3z=-16

-3w+3x-y+4z=20

5w+4x+3y-z=-10

-w+2x+5y+z=-4

Answer 1

Elimination method :

The system of equations are

   → ( 1 )

    → ( 2 )

   → ( 3 )

     → ( 4 )

To get the eq'ns (1) & (2) that contain opposite terms multiply the eq (2) by 2.

Write the equations in column form and add the equations to eliminate y - variable.

The resultant equation is    → ( 4 )

To get the eq'ns (2) & (3) that contain opposite terms multiply the eq (2) by 3.

Write the equations in column form and add the equations to eliminate y - variable.

The resultant equation is    → ( 5 )

To get the eq'ns (3) & (4) that contain opposite terms multiply the eq (3) by 5 and eq (4) by 3.

Write the equations in column form and Subtract the equations to eliminate y - variable.

The resultant equation is    → ( 6 )

Answer 2

To get the eq'ns (4) & (5) that contain opposite terms multiply the eq (4) by 2.

Write the equations in column form and subtract the equations to eliminate w - variable.

The resultant equation is    → ( 7 )

To get the eq'ns (4) & (6) that contain opposite terms multiply the eq (4) by 7.

Write the equations in column form and subtract the equations to eliminate w - variable.

The resultant equation is    → ( 8 )

To get the eq'ns (7) & (8) that contain opposite terms multiply the eq (8) by 31.

Write the equations in column form and add the equations to eliminate z - variable.

The resultant equation is  .

Substitute the value of in eq (7), and solve for z.

.

Substitute the values of and in eq (4), and solve for w.

.

Substitute the values of , , and in eq (1), and solve for y.

.

Therefore, solution of the given system is ,  , , and .