Solving a System of Equations Using Gaussian Elimination

To solve this system of equations, we can use Gaussian elimination. First, we can add equations (1) and (3) to eliminate u1:

2u2 - 3v1 + v2 - 2v4 = q

Next, we can add equations (2) and (5) to eliminate u1 again:

-8v1 - v2 - 9v4 = p

Now we have two equations in three variables (u2, v1, and v4). We can solve for u2 in terms of v1 and v4 using equation (4):

u2 = (1/2)v4 - 2v2 + (1/2)v1

Substituting this into the first equation, we get:

v4 - 6v2 - 3v1 = q/2

We can rearrange this equation to solve for v4:

v4 = 6v2 + 3v1 + q/2

Finally, we can substitute this expression for v4 into the second equation:

-8v1 - v2 - 9(6v2 + 3v1 + q/2) = p

Simplifying and solving for v2 in terms of v1 and q, we get:

v2 = (17/12)v1 - (1/24)q - (1/9)p

We can then substitute this expression for v2 into the equation for u2:

u2 = (1/2)(6v2 + 3v1 + q/2) - 2v2 + (1/2)v1

Simplifying, we get:

u2 = (1/4)v1 - (1/8)q - (1/4)p

Therefore, the solutions for u2, v1, and v4 in terms of p and q are:

u2 = (1/4)v1 - (1/8)q - (1/4)p

v4 = 6v2 + 3v1 + q/2 = 6[(17/12)v1 - (1/24)q - (1/9)p] + 3v1 + q/2 = (11/4)v1 - (1/12)q - (1/3)p

v2 = (17/12)v1 - (1/24)q - (1/9)p