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# Note for MATH 1313 with Professor Ahmed-Zaid at UH section 2.2

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Date Created: 02/06/15
9 2 gimme as Section 22 Solving Systems of Linear Equations I As you may recall from College Algebra you can solve a system of linear equations in two variables easily by applying the substitution or addition method Since these methods become tedious when solving a large system 0 uations a suitable technique for solving such systems of linear equations of any size is the This method involves a sequence of operations on a syste inear equatr ns to obt n at each stage an equivalent system The GaussJordan elimination method is complete when the original system has been transformed so that it is in a certain standard form from which the solution can be easily read Augmented Matrices Matrices are rectangular arrays of numbers We will use these to solve systems of equation in 2 or more variables LN QP L 2x4y6z22 Example Given 3x 8y 52 27 7 x y 2z 2 i 1 Z q 2 4 6 l 2 4 6 22 The coef cient matrix is 3 8 The augmented matrix is L 3 8 5 27 kcgl 2 14122 5 0 K71 A K 3 0v E 0v 39 71 i Q An augmented matrix is inMif it satis es the following conditions LAM wquot 1 Each row consisting entirely of zeros lies below other row having nonzero entries 2 The rst nonzero entry in each row is 1 called a 3 In any two successive nonzero rows the leading l in e ower row lies to the right of the leading l in the upper ro 4 If a column contains m then the other entries in Unit Column A column in a coef cient matrix is in unit form if one of the entries in the column is a l and the other entries are zeros Example 1 Determine which of the following matrices are in rowreduced form If a matrix is not in rowreduced form state which condition is violated l 0 a 0 l 0 0 Section 22 7 Solving Systems of Linear Equation I l 41 km W32 v NW FUJWWUK 6r a Kb K0003 ix L 9 A LLQR A h i 2 L m a 0 r5 3 21 m1quot 1 6 1amp1 1L 6q2KL 043 gt o sgt a 03913 1 0 d K I 9 MAW 1 millL L Q FQ L I x a amid Af l 171 392 a 21 3 s S tS K39L A o 0 1 Example 3 Solve the syste of linear equations using the GaussJordan elimination method 2xy72z4 x3yiz73 2 3x4yiz7 12 0 2 l 010 i D 6 3 3 pool 6Q Section 22 7 Solving Systems of Linear Equationl L3 422 amp393 010 11 11 f39LT 39i 7 r2 IL 39439 f Zie t 3 L 3 quot4 7 1 3 if i 7 14 1 l a l 6 AZ 392 Pg 9 o s o N w 7 Q 4 1 16 l 13 01 069 19 4 13 1 04314 23a 3 3 1 quot1 I o S 1 396 CD 0 9 Q 10 13 5721 G o R G Example 4 Solve the system of linear equations using the GaussJordan elimination method y 82 9 x 2 y 32 3 7y 52 12 Section 22 7 Solving Systems of Linear Equation I

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