Question

Find the general solution of the system whose augmented matrix is: [1, -2, -1, 3; 3, -6, -2, 2]

Answer 1

To find the general solution of the given system, we reduced the augmented matrix to row-echelon form, solved for z, and then expressed x in terms of a free variable y, leading to a parametric representation of the solution set.

Step-by-step explanation:

To find the general solution of the system represented by the augmented matrix [1, -2, -1, 3; 3, -6, -2, 2], we need to use Gaussian elimination or Gauss-Jordan elimination to reduce the matrix to its row-echelon form or reduced row-echelon form.

Let us start with the matrix:

• Row 1: [1, -2, -1, 3]
• Row 2: [3, -6, -2, 2]

We can multiply the first row by 3 and subtract it from the second row, which will give us:

• Row 1: [1, -2, -1, 3]
• Row 2: [0, 0, 1, -7]

We observe that Row 2 can be written as 0x + 0y + 1z = -7, meaning z = -7. Since Row 1 has three unknowns, let's express it as x - 2y - z = 3. Using our found value of z, we can substitute -7 for z and end up with x - 2y + 7 = 3, which simplifies to x - 2y = -4.

Since we have a free variable y, we can express the general solution in terms of y:

1. x = 2y - 4
2. y is free, we can choose any value for it
3. z = -7

The general solution will, therefore, involve a parameter (let's say λ for y) and will look like:

x = 2λ - 4
y = λ
z = -7