Question

Solve the following system(s) of linear equations by Gauss-Jordan Elimination method.

10x+y+z=12
x+10−z=10
x−2y+10z=9

Answer 1

To solve the system of linear equations by Gauss-Jordan Elimination, we convert the equations into an augmented matrix, perform row operations to reach reduced row-echelon form, and interpret the resulting matrix as the solution to the system.

Step-by-step explanation:

To solve the given system of linear equations using the Gauss-Jordan Elimination method, we first need to express the equations in augmented matrix form:

1. 10x + y + z = 12
2. x + 10y - z = 10
3. x - 2y + 10z = 9

Our augmented matrix will look like this:

[10 1 1 | 12]
[1 10 -1 | 10]
[1 -2 10 | 9]

We perform row operations to transform this matrix into reduced row-echelon form (RREF), which will give us the solution to the system. The goal is to have a matrix that has the form of an identity matrix on the left side and the solutions on the right:

For example, after using Gauss-Jordan elimination steps, we could end up with a matrix like this:

[1 0 0 | a]
[0 1 0 | b]
[0 0 1 | c]

Here, 'a,' 'b,' and 'c' would represent the values of x, y, and z respectively.

Solve the simultaneous equations by performing a sequence of operations such as swapping rows, multiplying rows by a non-zero scalar, and adding or subtracting multiples of rows from each other until an RREF is achieved. This process often requires careful checking and rechecking.