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01-linear - LINEAR EQUATIONS Math 21b O Knill RIDDLES"15...

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9/27/2004, LINEAR EQUATIONS Math 21b, O. Knill HOMEWORK due 9/29/2004: 1.1: 10,12,20,24,38b,26*,36* (Problems with * are recommended only). SYSTEM OF LINEAR EQUATIONS. A collection of linear equations is called a system of linear equations . An example is 3 x - y - z = 0 - x + 2 y - z = 0 - x - y + 3 z = 9 . It consists of three equations for three unknowns x, y, z . Linear means that no nonlinear terms like x 2 , x 3 , xy, yz 3 , sin( x ) etc. appear. A formal definition of linearity will be given later. LINEAR EQUATION. The equation ax + by = c is the general linear equation in two variables and ax + by + cz = d is the general linear equation in three variables. The general linear equation in n variables has the form a 1 x 1 + a 2 x 2 + ... + a n x n = 0 . Finitely many such equations form a system of linear equations . SOLVING BY ELIMINATION. Eliminate variables. In the first example, the first equation gives z = 3 x - y . Putting this into the second and third equation gives - x + 2 y - (3 x - y ) = 0 - x - y + 3(3 x - y ) = 9 or - 4 x + 3 y = 0 8 x - 4 y = 9 . The first equation gives y = 4 / 3 x and plugging this into the other equation gives 8 x - 16 / 3 x = 9 or 8 x = 27 which means x = 27 / 8. The other values y = 9 / 2 , z = 45 / 8 can now be obtained. SOLVE BY SUITABLE SUBTRACTION.
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