Unformatted text preview: pieces of information - i.e., two equations. Having more than two equations suggests we have more
information than necessary to determine the values of the unknowns. While this is not necessarily the case, it does
explain the choice of terminology ‘overdetermined’.
We need more than two variables to give an example of the latter.
Again, experience with systems with more variables helps to see this here, as does a solid course in Linear Algebra.
That is, a system with the same solution set. 8.1 Systems of Linear Equations: Gaussian Elimination 455 Theorem 8.1. Given a system of equations, the following moves will result in an equivalent
system of equations.
• Interchange the position of any two equations.
• Replace an equation with a nonzero multiple of itself.a
• Replace an equation with itself plus a nonzero multiple of another equation.
a That is, an equation which results from multiplying both sides of the equation by the same nonzero number. We have seen plenty of instances of the second and third moves in Theorem 8.1 when we solved
the systems Example 8.1.1. The ﬁrst move, while it obviously admits an equivalent system, seems
silly. Our perception will change as we consider more equations a...
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