Question

Solve the following system of linear equations by Gaussian elimination and back substitution. Express the answer as vectors or as a linear combination of vectors if appropriate.

a. 2x - y + z = 3
b. 4x - y - 2z = 7

Answer 1

The solution to the system of linear equations is obtained by Gaussian elimination, resulting in y = 1, and expressing z in terms of x, leading to a free variable. The solution is presented as a vector V = tI + 1J + (4 - 2t)K, where t is the free variable.

Step-by-step explanation:

To solve the system of linear equations by Gaussian elimination and back substitution, we have:

First, we use Gaussian elimination to reduce the system to row echelon form. We can multiply equation (a) by 2 and subtract it from equation (b) to eliminate x:

With y determined, we can substitute it back into either equation (a) or (b) to find z and x respectively. For simplicity, let's substitute y into equation (a):

Since there are only two equations and three unknowns, we have a free variable. We can express the solution as a linear combination of vectors with one of the variables being free (let's choose x as the free variable, t).

The solution to the system can be expressed as the vector:

V = tI + 1J + (4 - 2t)K