Final answer:
The solution set of the system x + y + u + z = 0 is found using free parameters for y, u, and z. The solution for x is given by -y - u - z, resulting in infinitely many solutions. Example values can be substituted back into the equation to verify the correctness of the solutions.
Step-by-step explanation:
To find the full solution set of the system given by the equation x + y + u + z = 0, we can apply Gaussian elimination. However, this equation represents a hyperplane in four-dimensional space and is underdetermined, meaning it has infinitely many solutions. The solution set can be described using free parameters.
Let's assign free variables to y, u, and z, then express x in terms of these variables:
This gives us the generic solution:
- x = -y - u - z
- y = y
- u = u
- z = z
We can choose any values for y, u, and z, and compute x to get a valid point in the solution set.
Example: If we let y = 1, u = -2, and z = 3, then x would be -1 - (-2) - 3 = -2. Thus, one of the solutions is (-2, 1, -2, 3).
To check the reasonableness of the solution, we can substitute the values back into the original equation to ensure that they satisfy x + y + u + z = 0.