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Unformatted text preview: How to Solve Linear Equations We give an algorithm to find out if a set of linear equations has a solution, and, if it does have a solution, how to find all of the solutions. Step One: First we rewrite the set of linear equations dropping much of the redundant information. If we have the equations x- 2 y + 3 z = 4 y- 7 z = 14 2 x- z = 0 , then we write this as 1- 2 3 4 1- 7 14 2- 1 . Note that we separate out the coefficients from the constants. We call such an array an augmented matrix. The sugmented part is the last column. A matrix is just a rectangular array of numbers with none of the columns or rows set out. For example 3 4- 6 23- 7 is a matrix. Step Two: To (try) to solve this set of equations we perform special operations on the equations, or rather we perform equivalent operations on the augmented matrix. These operations do not change the set of solutions to our initial equations. We can do three kinds of operations on the equations that do not change the set of solutions: 1. We can add a multiple of an equation to any other different equation....
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