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x + 2y - 2z = -2
\n" ); document.write( "5x + 9y -4z = -3
\n" ); document.write( "3x + 4y -5z = -3
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\n" ); document.write( "you want to get the equations in a form such that the 0 terms and the non zero terms form 2 triangles facing each other as part of a rectangle. the leading non-zero terms form a diagonal.
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\n" ); document.write( "that would be something like this.
\n" ); document.write( "ax.....by.....cz.....d
\n" ); document.write( "0......by.....cz.....d
\n" ); document.write( "0......0......cz.....d
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\n" ); document.write( "you can perform 3 basic operations on each equation.
\n" ); document.write( "1. you can interchange any rows with each other.
\n" ); document.write( "2. you can multiply any row by any constant.
\n" ); document.write( "3. you can take any row and multiply it by a constant, and then add to it, or subtract from it, any other row multiplied by a constant.
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\n" ); document.write( "you do this methodically working from top to bottom left to right.
\n" ); document.write( "get a nonzero in the first row of the first column.
\n" ); document.write( "then manipulate the equations so that all other rows in that column = 0.
\n" ); document.write( "get a nonzero in the second row of the second column.
\n" ); document.write( "then manipulate the equations so that all other rows below that column = 0.
\n" ); document.write( "repeat the procedure for all remaining rows and columns.
\n" ); document.write( "once you're done, you substitute your values into each equation to make sure that all equations have been satisfied.
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\n" ); document.write( "back go your equation:
\n" ); document.write( "x + 2y - 2z = -2
\n" ); document.write( "5x + 9y -4z = -3
\n" ); document.write( "3x + 4y -5z = -3
\n" ); document.write( "-----
\n" ); document.write( "since you have a nonzero in the first column first row then we'll leave that row intact for now.
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\n" ); document.write( "if you take the first row and multiply it by 5 and subtract it from the second row then the second row will become:
\n" ); document.write( "first row multiplied by 5: 5x + 10y - 10z = -10
\n" ); document.write( "second row before modification: 5x + 9y -4z = -3
\n" ); document.write( "second row after subtracting modified first row from it: 0x -y +6z = 7
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\n" ); document.write( "if you take the first row and multiply it by 3 and subtract it from the third row then the third row will become:
\n" ); document.write( "first row multiplied by 3: 3x + 6y - 6z = -6
\n" ); document.write( "third row before modification: 3x + 4y -5z = -3
\n" ); document.write( "third row after subtracting modified first row from it: 0x -2y + z = 3
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\n" ); document.write( "you now have 3 equations in which the first column is in the form you want it to be in:
\n" ); document.write( "those are:
\n" ); document.write( "x + 2y - 2z = -2
\n" ); document.write( "0x -y +6z = 7
\n" ); document.write( "0x -2y + z = 3
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\n" ); document.write( "next you work on the second column.
\n" ); document.write( "in this column you want the third row to be 0.
\n" ); document.write( "you could add the first row to the third row but then the first column in the third row would be nonzero so you don't want to do that.
\n" ); document.write( "work with the second and third rows only.
\n" ); document.write( "multiply the second row by 2 and subtract it from the third fow.
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\n" ); document.write( "second row multipled by 2: 0x -2y + 12z = 14
\n" ); document.write( "third row before modification: 0x -2y + z = 3
\n" ); document.write( "third row after subtracting modified second row from it: 0x + 0y -11z = -11
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\n" ); document.write( "you now have 3 equations with the first column and the second column in the form that you want them.
\n" ); document.write( "those are:
\n" ); document.write( "x + 2y - 2z = -2
\n" ); document.write( "0x -y +6z = 7
\n" ); document.write( "0x + 0y -11z = -11
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\n" ); document.write( "this is the simple gaussian elimination.
\n" ); document.write( "you can go one step further and make all the leading coefficients 1.
\n" ); document.write( "that i think is what they call the reduced form.
\n" ); document.write( "actually your equation is already in that form except for the third row, since the first row x coefficient is 1 and the second row y coefficient is -1.
\n" ); document.write( "to make the third row z coefficient 1, you would divided that equation by 11.
\n" ); document.write( "it would become 0x + 0y -z = -1
\n" ); document.write( "whether you do that or not you will get the same answer.
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\n" ); document.write( "working with the final modified equations and solving for x,y,z follows:
\n" ); document.write( "x + 2y - 2z = -2
\n" ); document.write( "0x -y +6z = 7
\n" ); document.write( "0x + 0y -11z = -11
\n" ); document.write( "-----
\n" ); document.write( "work from bottom and solve for z.
\n" ); document.write( "-11z = -11
\n" ); document.write( "z = 1
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\n" ); document.write( "go up one equation and solve for y, using the value of z already found.
\n" ); document.write( "-y + 6 = 7
\n" ); document.write( "-y = 1
\n" ); document.write( "y = -1
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\n" ); document.write( "go up one equation and solve for x, using the value of y and z already found.
\n" ); document.write( "x -2 -2 = -2
\n" ); document.write( "x -4 = -2
\n" ); document.write( "x = 2
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\n" ); document.write( "looks like the values you want are:
\n" ); document.write( "x = 2, y = -1, z = 1
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\n" ); document.write( "substitute those values in each of the 3 original equations you started with.
\n" ); document.write( "3 original equations:
\n" ); document.write( "x + 2y - 2z = -2
\n" ); document.write( "5x + 9y -4z = -3
\n" ); document.write( "3x + 4y -5z = -3
\n" ); document.write( "-----
\n" ); document.write( "solve first row equation using x = 2, y = -1, z = 1:
\n" ); document.write( "x + 2y - 2z = -2
\n" ); document.write( "2 -2 -2 = -2
\n" ); document.write( "-2 = -2
\n" ); document.write( "first equation looks good.
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\n" ); document.write( "solve second row equation using x = 2, y = -1, z = 1:
\n" ); document.write( "5x + 9y -4z = -3
\n" ); document.write( "10 -9 -4 = -3
\n" ); document.write( "1 - 4 = -3
\n" ); document.write( "-3 = -3
\n" ); document.write( "second equation looks good.
\n" ); document.write( "-----
\n" ); document.write( "solve third row equation using x = 2, y = -1, z = 1:
\n" ); document.write( "3x + 4y -5z = -3
\n" ); document.write( "6 -4 -5 = -3
\n" ); document.write( "2 - 5 = -3
\n" ); document.write( "-3 = -3
\n" ); document.write( "third equation looks good.
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