help me find the number of solutions of the given linear equation system

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x1+ 2 x2+ 3 x3+4 x4= 7
2 x1+ x2+2 x3 +3 x4=6
3 x1 +2 x2+x3 +2 x4= 7
4 x1 +3 x2+ a x3+ x4=a

Accepted Answer

Walter Roberson
Walter Roberson on 25 Nov 2021
The number of solutions to a set of linear equations is always exactly zero, or exactly one, or infinite.
The number is zero when the system is inconsistent, which requires more equations than variables. That cannot happen in your system.
The number would be infinite if you can make the 4th equation be redundant. That is potentially possible if the "a" value is a multiple of one of the x variables. You can work through to figure out what values would introduce a singularity.
  1 Comment
Walter Roberson
Walter Roberson on 25 Nov 2021
ah... suppose that you make "a" equal to the expression at the left of equation 4, plus 1. Then that equation would be inconsistent, so you can get the case of 0 solutions.

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