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**Unformatted text preview: **Math 313 Lecture #1 Â§ 1.1: Systems of Linear Equations Consistency and Inconsistency of Systems. An m Ã— n system of linear equa- tions (or simply system ) is m linear equations in n unknowns: a 11 x 1 + a 12 x 2 + Â· Â· Â· + a 1 n x n = b 1 , a 21 x 1 + a 22 x 2 + Â· Â· Â· + a 2 n x n = b 2 , . . . . . . . . . . . . . . . a m 1 x 1 + a m 2 x 2 + Â· Â· Â· + a mn x n = b m , where the a ij â€™s and the b i â€™s are all real numbers, and the x i â€™s are the variables. A solution of an m Ã— n system is an ordered n-tuple of numbers ( x 1 , x 2 , . . . , x n ) that satisfy all m equations simultaneously. Example. Consider the three 2 Ã— 2 systems: 2 x 1 + 4 x 2 = 6 , x 1 + 2 x 2 = 5 , 2 x 1 + x 2 = 1 , x 1 + x 2 =- 1 , x 1- 2 x 2 = 3 , 2 x 1- 4 x 2 = 6 . Each linear equation here can be represented as a line in the plane: x2 4 4 2 2 ! 2 ! 4 x1 5 5 3 1 3 ! 1 ! 3 1 ! 5 ! 1 ! 2 ! 3 ! 4 ! 5 x2 4 x1 2 3 ! 2 1 ! 4 ! 1 ! 3 5 3 1 4 ! 1 ! 3 2 ! 5 ! 2 ! 5 ! 4 5 ! 1 3 4 x2 ! 2 ! 5 3 ! 2 5 4 ! 4 1 ! 3 ! 1 ! 3 x1 ! 5 2 ! 4 1 2 The ordered pair (2 ,- 3) is a solution for the second system. [Check it X .] And there are no other solutions. The ordered pair (3 , 0) is a solution for the third system. Actually, an ordered pair of the form (3 + 2 t, t ) for any real t is a solution for the third system. [Check them X .] However, the first systems has no solution. Why? Because if the ordered pair ( x 1 , x 2 ) is a solution, then x 1 = 5- 2 x 2 and 2 x 1 = 6- 4 x 2 ; but the second of these can be written as x 1 = 3- 2 x 2 , so that 5- 2 x 2 = x 1 = 3- 2 x 2 which implies that 5 = 3. / / / / The solution set of an m Ã— n system is the collection of all of its solutions....

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