Solve each system of three linear equations containing three unknowns.

Question

x-3y+z=13

3x+y-4z=13

-4x-4y+2z=0

Answer 1

The system of equations are .

Use the elimination method to make a system of two equations in two variables.

To get two equations 1 and 2 that contain opposite coefficient of z - variable multiply the first equation by 4.

Write the equations 1 and 2 in column form and add the corresponding columns to eliminate z - variable.

The resultant equation is taken as fourth equation : 7x - 11y = 65.

To get two equations 2 and 3 that contain opposite coefficient of z - variable multiply the third equation by 2.

Write the equations 2 and 3 in column form and add the corresponding columns to eliminate z - variable.

The resultant equation is taken as fifth equation : - 5x - 7y = 13.

Solve the system of two equations with two variables.

To get two equations 4 and 5 that contain opposite coefficient of x - variable multiply the fourth equation by 5 and multiply the fifth equation by 7.

Write the equations 4 and 5 in column form and add the corresponding columns to eliminate x - variable.

The resultant equation - 104y = 416 ------> y = - 4.

Use one of the equation with two variables (Equation: 4 or 5) to solve for x.

The fourth equation: 7x - 11y = 65.

7x - 11(- 4) = 65

7x + 44 = 65

7x = 21

x = 3.

Solve for z using one of the original equations with three variables.

The first equation: x - 3y + z = 13.

(3) - 3(- 4) + z = 13

3 + 12 + z = 13

15 + z = 13

z = - 2.

The solution (x, y, z) = (3, - 4, - 2).