Section 1.1: Systems of Linear Equations A linear equation: a1x1 ...

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Two Fundamental Questions (Existence and Uniqueness)1) Is the system consistent; (i.e. does a solution exist?)2) If a solution exists, is it unique? (i.e. is there one & only onesolution?)EXAMPLE: Is this system consistent?x 1 2x 2 x 3 02x 2 8x 3 84x 1 5x 2 9x 3 9In the last example, this system was reduced to the triangularform:x 1 2x 2 x 3 0x 2 4x 3 4x 3 31 2 1 00 1 4 40 0 1 3This is sufficient to see that the system is consistent and unique.Why?10
EXAMPLE: Is this system consistent?3x 2 6x 3 8x 1 2x 2 3x 3 15x 1 7x 2 9x 3 00 3 6 81 2 3 15 7 9 0Solution: Row operations produce:0 3 6 81 2 3 11 2 3 10 3 6 81 2 3 10 3 6 85 7 9 00 3 6 50 0 0 3Equation notation <strong>of</strong> triangular form:x 1 2x 2 3x 3 13x 2 6x 3 80x 3 3 Never trueThe original system is inconsistent!11
Page 1 and 2: Section 1.1: Systems of Linear Equa
Page 3 and 4: EXAMPLE: Three equations in three v
Page 5 and 6: x 2 x 2−10 −5 5 10 x 14422−10
Page 7 and 8: Elementary Row Operations:1. (Repla
Page 9: x 1 29x 2 16x 3 31 0 0 290 1 0 1

Section 1.1: Systems of Linear Equations A linear equation: a1x1 ...

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