If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent.
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This statement is not necessarily true.
An inconsistent system of linear equations is one where there are no solutions that satisfy all the equations in the system. For example, the system:
x + y = 3
x – y = 5
2x – 2y = 6
is inconsistent because if we add the first two equations together, we get:
2x = 8
which implies that x = 4. But if we substitute x = 4 into the first equation, we get:
4 + y = 3
which is not possible for any real value of y. So there is no solution to this system of equations.
However, if the number of equations in a linear system exceeds the number of unknowns, it does not necessarily mean that the system is inconsistent. For example, the system:
x + y + z = 6
2x – y + 3z = 0
x – 2y + 5z = -4
4x + y – z = 8
has four equations and three unknowns. But it is consistent because there is a unique solution to the system, which can be found using techniques such as Gaussian elimination or matrix inversion.
So the statement If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent is false.
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