document.write( "Question 1129997: Could someone assist in explaining how to correctly solve the question. My previous answer was -40-x-3z/2.\r
\n" ); document.write( "\n" ); document.write( "Solve the system by Gaussian elimination. (If there is no solution, enter NO SOLUTION. If there are an infinite number of solutions, enter a general solution in terms of one of the variables.)
\n" ); document.write( "0.1x − 0.2y + 0.3z = 4
\n" ); document.write( "0.5x − 0.1y + 0.4z = 16
\n" ); document.write( "0.7x − 0.2y + 0.3z = 16\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #746648 by greenestamps(13200)\"\" \"About 
You can put this solution on YOUR website!


\n" ); document.write( "Your \"answer\" makes no sense; the answer should show the values of x, y, and z that satisfy all three equations.

\n" ); document.write( "There are always an endless number of different ways to solve a system of equations using Gaussian elimination. I will show one.

\n" ); document.write( "The beginning matrix is this:

\n" ); document.write( "\"matrix%283%2C4%2C0.1%2C-0.2%2C0.3%2C4%2C0.5%2C-0.1%2C0.4%2C16%2C0.7%2C-0.2%2C0.3%2C16%29\"

\n" ); document.write( "The first thing I would do is multiply each row (i.e., each equation) by 10 to get rid of the decimals, since working with integers is much easier.

\n" ); document.write( "\"matrix%283%2C4%2C1%2C-2%2C3%2C40%2C5%2C-1%2C4%2C160%2C7%2C-2%2C3%2C160%29\"

\n" ); document.write( "Now before I plunge into the standard techniques for Gaussian elimination, I'm going to look at the equations to see if there is some major simplification that can be made. And I see that the coefficients of y and z in both the first and third equations are the same (-2 and 3); that means I can get an equation in x only by subtracting the first equation from the third.

\n" ); document.write( "So my first step (after converting the matrix to all integers) will be to replace row 3 with (row 3 minus row 1):

\n" ); document.write( "\"matrix%283%2C4%2C1%2C-2%2C3%2C40%2C5%2C-1%2C4%2C160%2C6%2C0%2C0%2C120%29\"

\n" ); document.write( "Next I will combine two steps: dividing row 3 by 6, and making it the first row (since it will be in the form I want for the first row of the final matrix):

\n" ); document.write( "\"matrix%283%2C4%2C1%2C0%2C0%2C20%2C1%2C-2%2C3%2C40%2C5%2C-1%2C4%2C160%29\"

\n" ); document.write( "Next use row 1 to get 0's in the first column of rows 2 and 3: replace row 2 with (row 1 minus row 2) and replace row 3 with (row 3 minus 5 times row 1):

\n" ); document.write( "\"matrix%283%2C4%2C1%2C0%2C0%2C20%2C0%2C2%2C-3%2C-20%2C0%2C-1%2C4%2C60%29\"

\n" ); document.write( "The first column of my matrix is in its final form; next I want to get a \"1\" in row 2 column 2. I can do that by replacing row 2 with (row 2 plus row 3):

\n" ); document.write( "\"matrix%283%2C4%2C1%2C0%2C0%2C20%2C0%2C1%2C1%2C40%2C0%2C-1%2C4%2C60%29\"

\n" ); document.write( "Next, to finish column 2, I need a \"0\" in row 3 column 2; I can do that by replacing row 3 with (row 3 plus row 2):

\n" ); document.write( "\"matrix%283%2C4%2C1%2C0%2C0%2C20%2C0%2C1%2C1%2C40%2C0%2C0%2C5%2C100%29\"

\n" ); document.write( "Column 2 is finished; next I need a \"1\" in row 3 column 3. I can get that by dividing row 3 by 5:

\n" ); document.write( "\"matrix%283%2C4%2C1%2C0%2C0%2C20%2C0%2C1%2C1%2C40%2C0%2C0%2C1%2C20%29\"

\n" ); document.write( "And last I need a \"0\" in row 2 column 3; I get that by replacing row 2 with (row 2 minus row 3):

\n" ); document.write( "\"matrix%283%2C4%2C1%2C0%2C0%2C20%2C0%2C1%2C0%2C20%2C0%2C0%2C1%2C20%29\"

\n" ); document.write( "The matrix is in the desired final form; it shows the solution to the system is

\n" ); document.write( "x=20; y=20; z=20 \n" ); document.write( "
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