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SOLVING SYSTEMS OF LINEAR EQUATIONS An equation is said to be linear if every variable has degree equal to one (or zero) is a linear equation is NOT a linear equation

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Review these familiar techniques for solving 2 equations in 2 variables. The same techniques will be extended to accommodate larger systems. Times 3 Add Substitute to solve: y=1

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L 1 represents line one L 2 represents line two L 1 is replaced by 3L 1 L 1 is replaced by L 1 + L 2 L 1 is replaced by ( 1/7)L 1

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These systems are said to be EQUIVALENT because they have the SAME SOLUTION.

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PERFORM ANY OF THESE OPERATIONS ON A SYSTEM OF LINEAR EQUATIONS TO PRODUCE AN EQUIVALENT SYSTEM: INTERCHANGE two equations (or lines) REPLACE L n with k L n, k is NOT ZERO REPLACE L n with L n + c L m note: L n is always part of what replaces it.

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EXAMPLES: is equivalent to L1L2L3L1L2L3 L1L3L2L1L3L2 L1L2L3L1L2L3 L14L2L3L14L2L3 L 1 L 2 L 3 + 2L 1 L1L2L3L1L2L3

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1 z 343zyx042 zyx Replace L 1 with (1/2) L 1 Replace L 3 with L 3 + L 2 Replace L 2 with L 2 + L 1 1 z 3y042 zyx 1 3 z y6 x +4z Replace L 1 with L 1 + 2 L 2

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Replace L 1 with L 1 + - 4 L 3

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1 3 z y2 x is EQUIVALENT to

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To solve the following system, we look for an equivalent system whose solution is more obvious. In the process, we manipulate only the numerical coefficients, and it is not necessary to rewrite variable symbols and equal signs:

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This rectangular arrangement of numbers is called a MATRIX

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2 Replace L 1 with L 1 + 2 L 2

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1 Replace L 3 with L 3 + L 2

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1 Replace L 2 with L 2 + 1L 1

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Interchange L 2 and L 3

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Replace L 1 with L 1 + -1 L 2 Replace L 3 with -1 L 3 Replace L 2 with -1 L 2

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The original matrix represents a system that is equivalent to this final matrix whose solution is obvious

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The original matrix represents a system that is equivalent to this final matrix whose solution is obvious

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The zeros The diagonal of ones Note the format of the matrix that yields this obvious solution: Whenever possible, aim for this format.

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8.1 Matrices and Systems of Equations. Let’s do another one: we’ll keep this one Now we’ll use the 2 equations we have with y and z to eliminate the y’s.

8.1 Matrices and Systems of Equations. Let’s do another one: we’ll keep this one Now we’ll use the 2 equations we have with y and z to eliminate the y’s.

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