Question

Consider the linear system x+y+2z+3w=13; x-2y+z+w=8; 3x+y+z-w=1.

A) Find all solutions (if any) by using the Gaussian elimination method.
B) Find all solutions(if any) by using the Gaussian elimination method.

Answer 1

To solve the given linear system using Gaussian elimination method, we can follow the steps of row operations to transform the matrix into row-echelon form and back-substitute to find the values of the variables. The solution to the given linear system is x = 11 1/3, y = 2/3, z = 1.

Step-by-step explanation:

To solve the given linear system using Gaussian elimination method:

Step 1: Write the augmented matrix for the system:

[1 1 2 3 13]

[1 -2 1 1 8]

[3 1 1 -1 1]

Step 2: Perform row operations to transform the matrix into row-echelon form:

R2 = R2 - R1, R3 = R3 - 3R1:

[1 1 2 3 13]

[0 -3 -1 -2 -5]

[0 -2 -5 -10 -38]

Step 3: Perform row operations to obtain row-echelon form:

R3 = R3 - (2/3)R2:

[1 1 2 3 13]

[0 -3 -1 -2 -5]

[0 0 -4/3 -4/3 -4]

Step 4: Back-substitute to find the values of the variables:

Substitute z = 1 into the second equation to find y:

-3y - 1 - 2 = -5 ⇒ -3y = -2 ⇒ y = 2/3

Substitute y = 2/3 and z = 1 into the first equation to find x:

x + 2/3 + 2 = 13 ⇒ x = 11 1/3

The solution to the given linear system is x = 11 1/3, y = 2/3, z = 1.