We can ask two basic questions about any equation, or system of equations: is there a solution, and, if there is, is it unique? Experience with finite systems of linear equations indicates that the two questions are interconnected. Consider for instance the equations

2*x* + 3*y* - 5*z* = *a*,

*x* - 2*y* + *z* = *b*,

3*x* +*y* - 4*z* = *c*.

Notice that the left-hand side of the third equation is the sum of the left-hand sides of the first two. As a result, no solution to the system exists unless *a* + *b* = *c*. But if *a* + *b* = *c*, then any solution of the first two equations is also a solution of the third; and in any linear system involving more unknowns than equations, solutions, when ...

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