Which one of the following is not a basic solution of system of linear equation

x1 + 2x2 + x3 = 4

2x1 + x2 + 5x3 = 5

The correct answer is Option 1.

Key Points

• To determine if a given set of values is a basic solution for a system of linear equations, we need to check if the values satisfy all the equations in the system.
• The given system of linear equations is:

                  x1 + 2x2 + x3 = 4
                  2x1 + x2 + 5x3 = 5
              

• Let's evaluate each option:
• Option 1: x1 = -1, x2 = 2, x3 = 1
  • Substitute into the first equation: -1 + 2(2) + 1 = -1 + 4 + 1 = 4 (satisfies)
  • Substitute into the second equation: 2(-1) + 2 + 5(1) = -2 + 2 + 5 = 5 (satisfies)
• Option 2: x1 = 2, x2 = 1
  • Missing x3 value, so this option cannot satisfy both equations simultaneously. This is the correct answer as it is incomplete.
• Option 3: x1 = 5, x3 = -1
  • Missing x2 value, so this option cannot satisfy both equations simultaneously.
• Option 4: x2 = 5/3, x3 = 2/3
  • Missing x1 value, so this option cannot satisfy both equations simultaneously.

Additional Information

• In a system of linear equations, a basic solution typically involves having the same number of independent equations as unknowns, with each unknown having a unique value.
• Basic solutions can be found using methods such as substitution, elimination, or matrix operations.
• Consistency of the system needs to be checked to ensure that there are no contradictions in the equations.