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34. Solving systems of linear equations

# 34. Solving systems of linear equations - Solving Systems...

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©2009 by L. Lagerstrom Solving Systems of Linear Equations For an introduction to the basic concepts, see the associated video clip Solution by matrix inversion Solution by matrix division Checking a solution exists Intersection of two lines

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Matlab code Command window display ©2009 by L. Lagerstrom Solution by Matrix Inversion %Consider the following 3 equations %in 3 unknowns: % 3*x1 + 2*x2 - x3 = 10 % -x1 + 3*x2 + 2*x3 = 5 % x1 - x2 - x3 = -1 %We write this in Ax = b form, where %A is the matrix of coefficients, %x is a column vector with the %elements x1, x2, and x3, and b is %a column vector with 10, 5, -1: A = [3 2 -1; -1 3 2; 1 -1 -1] b = [10; 5; -1] %Find x by getting the inverse of A %and multiplying it by b: x = inv(A)*b %Remember: Before writing down A and %b, all the terms in the 3 equations %must be lined up properly and all %the constants must be on the right %side of the equations. A = 3 2 -1 -1 3 2 1 -1 -1 b = 10 5 -1 x = -2.00 5.00 -6.00
Matlab code Command window display ©2009 by L. Lagerstrom Solution by Matrix Division %Alternatively, we can solve for x %by matrix division (the underlying %method uses a technique called %"Gauss elimination"): %Define A and b as before: A = [3 2 -1; -1 3 2; 1 -1 -1] b = [10; 5; -1] %Find x by using the left division %operator: x = A\b %This time we'll also confirm that %the values in x solve the equations %by plugging in x and seeing if %we get b (i.e., A*x should = b): result = A*x A = 3 2 -1 -1 3 2 1 -1 -1 b = 10 5 -1 x = -2.00 5.00 -6.00 result = 10.00 5.00 -1.00

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