linear equation and gaussian elimination

linear equation and gaussian elimination

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  Linear Systems & Gaussian Elimination AjuGeorge SCET KODAKARA
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  Contents o Linear Equation oBack substitution o Gaussian Elimination method o Solving of linear equation using Gaussian Elimination method o Some examples o Bibliography
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  What is linearequation? An equation contains variables that gives a straight line when plotted on a graph. Linear equations can have one or more variables. An example of a linear equation with three variables, x, y, and z, is given by: ax + by + cz + d = 0, where a, b, c, and d are constants and a, b, and c are non-zero
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  What is backsubstitution method? • The process of solving a linear system of equations that has been transformed into row- echelon form or reduced row-echelon form. The last equation is solved first, then the next- to-last, etc • For Example ; X-2y+z=4 Y+6z=-1 Z=2
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  Gaussian elimination method •Gaussian elimination (also known as row reduction) is an algorithm for solving systems of linear equations. • To perform row reduction on a matrix, one uses a sequence of elementary row operations to modify the matrix until the lower left-hand corner of the matrix is filled with zeros, as much as possible • Using these operations, a matrix can always be transformed into an upper triangular matrix, and in fact one that is in row echelon form.
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  For Example;
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  How to solve? Example1 x + 5y= 7-----(1) −2x − 7y = −5.-----(2)
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  Step 1 572 751        Make theEquation into matrix form Ie, x + 5y= 7-----(1) −2x − 7y = −5.-----(2) becomes
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  Step 2        572 751 930 751       Findsuitable elementary transformation method to form a upper triangular matrix In this problem “Add twice Row 1 to Row 2”
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        930 751 310 751       Multiply Row 2by 1/3. This matrix gives Y=3
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  By back substitution, We know that y=3 From equation (1) x + 5y= 7 X+5*3=7 X=7-15 X=-8
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  Some problems • UseGaussian elimination to solve the system of linear equations 2x2 + x3 = −8 x1 − 2x2 − 3x3 = 0 −x1 + x2 + 2x3 = 3 • Use Gaussian elimination to solve the system of linear equations x1 − 2x2 − 6x3 = 12 2x1 + 4x2 + 12x3 = −17 x1 − 4x2 − 12x3 = 22..
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  Bibilography • https ://en.wikipedia.org/wiki/Gaussian_elimination •https ://math.dartmouth.edu/archive/m23s06/public_htm l/handouts/row_reduction_examples.pdf • http ://www.purplemath.com/modules/systlin6.html