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Unformatted text preview: MA 265 LECTURE NOTES: WEDNESDAY, JANUARY 9 Two Variables General Remarks. In the previous lecture, we gave several examples of systems of m equations in n = 2 variables. More generally, say that we have a system of linear equations in the form a 11 x + a 12 y = b 1 a 21 x + a 22 y = b 2 As in the previous lecture, we can express these linear equations as lines: y = a 11 a 12 x + b 1 a 12 y = a 21 a 22 x + b 2 a 22 If these lines have different slopes, i.e., a 11 a 12 negationslash = a 21 a 22 = a 11 a 22 a 12 a 21 negationslash = 0 then we expect to find precisely one solution. That means the system is consistent. However, if the slopes are equal, i.e., a 11 a 12 = a 21 a 22 = a 11 a 22 a 12 a 21 = 0 then we have parallel lines. We must consider the intercepts in order to determine whether the lines are different (i.e., no solutions) or overlap (i.e., infinitely many solutions). Three Variables Example. We now consider the case where we have n = 3 variables. Consider the linear system x + 2 y + 3 z = 6 2 x 3 y + 2 z = 14 3 x +...
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 Spring '10
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