Question

Consider the linear system of equations Ax = b: 3X₁ - X₂ + X₃ = 3 X₁ + 3X₂ + X₃ =0 X₁ +X₂+3X₃=1 Put this system in matrix form Ax= b and then find the solution for x. Show the step-by-step working for solving this system.

Answer 1

To put the system of equations in matrix form, we rewrite it as an augmented matrix and perform row operations to transform it into row-echelon form. Then, we use back substitution to find the values of X₁, X₂, and X₃. The solution is X₁=1/7, X₂=-3/7, and X₃=8/7.

Step-by-step explanation:

To put the system of equations in matrix form, we can rewrite the equations as follows:

`3X₁ - X₂ + X₃ = 3`

`X₁ + 3X₂ + X₃ = 0`

`X₁ + X₂ + 3X₃ = 1`

This can be represented as:

| 3 -1 1 | | X₁ | | 3 |

| 1 3 1 | x | X₂ | = | 0 |

| 1 1 3 | | X₃ | | 1 |

To solve this system, we can use row operations to transform the augmented matrix into row-echelon form, and then use back substitution to find the values of X₁, X₂, and X₃.

The step-by-step process would involve:

1. Multiply the first row by 1/3 to make the leading coefficient of the first row equal to 1.
2. Add -1 times the first row to the second row to eliminate the X₁ term.
3. Add -1 times the first row to the third row to eliminate the X₁ term.
4. Multiply the second row by 1/4 to make the leading coefficient of the second row equal to 1.
5. Add -1/2 times the second row to the third row to eliminate the X₂ term.
6. Multiply the third row by 2/7 to make the leading coefficient of the third row equal to 1.
7. Back substitute to find the values of X₁, X₂, and X₃.

The solution to the system of equations is X₁ = 1/7, X₂=-3/7, and X₃=8/7.