Final answer:
To solve the given system of equations using Gaussian elimination, we convert it to an augmented matrix and apply row operations. We find that the system is consistent with an infinite number of solutions. By setting arbitrary values for two of the variables, we can obtain three particular solutions.
Step-by-step explanation:
To solve the simultaneous equations for the unknowns using Gaussian elimination, we first write the system in matrix form. The system is:
- 5x₁ - 5x₂ - 15x₃ - 3x₄ = -34
- -2x₁ + 2x₂ + 6x₃ + x₄ = 12
Transforming it into an augmented matrix, we get:
[5 -5 -15 -3 | -34]
[-2 2 6 1 | 12]
Using elementary row operations, we aim to get a matrix in reduced row-echelon form. We'll multiply the second row by -2.5 and add it to the first row to eliminate x₁.
After the operation, the first row becomes:
[0 0 -15 -5.5 | -4]
We notice that the system has one equation with three unknowns, suggesting that there might be an infinite number of solutions if the equation is consistent. Since the second row is not a contradiction and does not provide any additional information, we can say this system is consistent and has infinite solutions.
To specify three particular solutions, we can set arbitrary values for x₃ and x₄, then solve for x₂ using the first equation.
These could be, for example:
- x₃ = 0, x₄ = 0
- x₃ = 1, x₄ = -1
- x₃ = -1, x₄ = 1
Check the answer to ensure the solutions do indeed satisfy the original equations.