Solving Systems of Three EquationsDate: 05/07/98 at 20:26:26 From: Claire Subject: Solving linear systems 3x + 4y + 8z = 5 9x - 5y + 6z = 2 15x + 2y - 4z = 5 I tried solving for x, then y and the z but I still ended up with extra variables. Help! Date: 05/09/98 at 02:41:33 From: Doctor Alain Subject: Re: Solving linear systems This is a system of three linear equations in the three unknowns x, y, and z. Geometrically, each equation describes a plane in space. The solution is the one point of space common to the three planes. There are systematic ways of solving such systems. One is called the Gauss/Jordan algorithm. I will give a solution based on the Gauss/ Jordan algorithm but avoiding fractions. I will leave the symbols x, y and z in the calculations; however, these are normally left out in the Gauss/Jordan algorithm. First we want to have zeros in the first column under the first row. We do this by subtracting 3 times row 1 from row 2 and 5 times row 1 from row 3: 3x + 4y + 8z = 5 0x - 17y - 18z = -13 0x - 18y - 44z = -20 Then we want to put a zero in column 2 under row 2. We do this in two steps. First we multiply row 2 by -18 and row 3 by 17. We get: 3x + 4y + 8z = 5 0x + 306y + 324z = 234 0x - 306y - 748z = -340 Now we add row 2 to row 3: 3x + 4y + 8z = 5 0x + 306y + 324z = 234 0x + 0y - 424z = -106 Now you can find z with row 3. Once you find z, you can find y with row 2, and when that is done, you can find x using row 1 and the values of y and z. -Doctor Alain, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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