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## SOLVING SYSTEMS OF LINEAR EQUATIONS

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**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 SOLVING SYSTEMS OF LINEAR EQUATIONS**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**L1 is replaced by 3L1**L1 is replaced by L1 + L2 L1 is replaced by (1/7)L1 L1 represents line oneL2 represents line two**These systems are said to be EQUIVALENT**because they have the SAME SOLUTION.**PERFORM ANY OF THESE OPERATIONS ON A SYSTEM OF LINEAR**EQUATIONS TO PRODUCE AN EQUIVALENT SYSTEM: • INTERCHANGE two equations (or lines) • REPLACE Ln with k Ln , k is NOT ZERO • REPLACE Ln with Ln + cLm • note: Ln is always part of what replaces it.**L1**L3 L2 L1 L2 L3 is equivalent to L1 L2 L3 L1 4L2 L3 is equivalent to L1 L2 L3 L1 L2 L3 + 2L1 is equivalent to EXAMPLES:**-**+ = x 2 y 4 z 0 - + - = x 3 y 4 z 3 = z 1 - + = x 2 y 4 z 0 = y 3 = z 1 = x +4z 6 = y 3 = z 1 Replace L1 with (1/2) L1 Replace L3 with L3 + L2 Replace L2 with L2 + L1 Replace L1 with L1 + 2 L2**Replace L1 with**L1 + - 4 L3**is EQUIVALENT to**= x 2 = y 3 = z 1**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:**2**Replace L1 with L1 + 2 L2**1**Replace L3 with L3 + L2**Replace L1 with L1 + -1 L2**Replace L2 with -1 L2 Replace L3 with -1 L3**The original matrix represents**a system that is equivalent to this final matrix whose solution is obvious**The original matrix represents**a system that is equivalent to this final matrix whose solution is obvious**The diagonal of ones**The zeros Note the format of the matrix that yields this obvious solution: Whenever possible, aim for this format.