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Use the Gauss Jordan method to solve the system of equations:
\n" ); document.write( "x+y+2z=7.......... I
\n" ); document.write( "3x-y+z=10............II
\n" ); document.write( "2x+y-3z=-6................III
\n" ); document.write( "THE COEFFICIENT AND AUGMENTED MATRIX IS
\n" ); document.write( "X,Y,Z,CONSTANT
\n" ); document.write( "-------------------------------------------
\n" ); document.write( "1,1,2,7...............MEANS EQN.I
\n" ); document.write( "3,-1,1,10.............MEANS EQN.II
\n" ); document.write( "2,1,-3,-6.............MEANS EQN.III\r
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\n" ); document.write( "SEE THE FOLLOWING EXAMPLE SOLVED FOR A PROBLEM SIMILAR TO YOURS WHICH IS
\n" ); document.write( "ESSENTIALLY SAME BUT FOR THE NUMBERS , FOLLOW STEP BY
\n" ); document.write( "STEP AND DO BY YOUR SELF,THERE BY YOU WILL LEARN HOW
\n" ); document.write( "TO DO AND BECOME AN EXPERT BY YOUR SELF!!!!IF YOU DO
\n" ); document.write( "NOT UNDERSTAND ASK WHERE IT IS NOT CLEAR...
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\n" ); document.write( "\n" ); document.write( "Linear_Algebra/20750: Hi, I'm in homeschooling and I'm
\n" ); document.write( "having trouble with matrices. I was wondering how to
\n" ); document.write( "solve the problem where you have to find the x,y, and
\n" ); document.write( "z values in the matrix:
\n" ); document.write( "[7 -7 5 | 9]
\n" ); document.write( "[9 5 -7 | -17]
\n" ); document.write( "[6 1 -7 | -2]
\n" ); document.write( "I'd appreciate the help. Thank you!
\n" ); document.write( "Caitlyn Reese
\n" ); document.write( "1 solutions
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\n" ); document.write( "Answer 9969 by venugopalramana(390) on 2005-11-28
\n" ); document.write( "07:14:11 (Show Source):
\n" ); document.write( "the 4 column heads represent x,y,z and constant term
\n" ); document.write( "in the matrix of system of eqns.
\n" ); document.write( "then each row gives us one eqn.like say row 1 gives us
\n" ); document.write( "that 7x-7y+5z=9..etc…
\n" ); document.write( "hence if we can make the matrix to become
\n" ); document.write( "1 0 0 ?
\n" ); document.write( "0 1 0 ??
\n" ); document.write( "0 0 1 ???
\n" ); document.write( "then from the explantion given above it means
\n" ); document.write( "1x=?.1y=?? And 1z=???
\n" ); document.write( "so we try to transform the matrix in to that form..by
\n" ); document.write( "the following steps.
\n" ); document.write( "in fact using the above explanation,you can see that
\n" ); document.write( "what we do at each step is just
\n" ); document.write( "divide each eqn. with a constant/add/subtract etc
\n" ); document.write( "which does not change the basic
\n" ); document.write( "eqn.for ex. dividing row 1 by 7 means change the given
\n" ); document.write( "eqn.7x-7y+5z=9 to x-y+5z/7=9/7
\n" ); document.write( "legend:- or1 means old row 1..nr1 means new row 1…r1
\n" ); document.write( "means the existing row 1 please note that no changes
\n" ); document.write( "are made in rows other than those mentioned at each
\n" ); document.write( "step.
\n" ); document.write( "start with given matrix …\r
\n" ); document.write( "\n" ); document.write( "7...... -7..... 5...... 9
\n" ); document.write( "9...... 5...... -7..... -17
\n" ); document.write( "6...... 1...... -7..... -2
\n" ); document.write( "step 1…we want to make 1st.row 1st.column as
\n" ); document.write( "1….so….nr1=or1/7...
\n" ); document.write( "1...... -1..... (5/7).. (9/7)
\n" ); document.write( "9...... 5...... -7..... -17
\n" ); document.write( "6...... 1...... -7..... -2
\n" ); document.write( "step 2..we want to make 2nd/3rd.rows,col.1 as
\n" ); document.write( "0...so...nr2=or2-9*r1........nr3=or3-6*r1
\n" ); document.write( "1...... -1..... (5/7).. (9/7)
\n" ); document.write( "0... 14..... (-7-9*5/7).... (-17-9*9/7)
\n" ); document.write( "0...... 7...... (-7-6*5/7)..... (-2-6*9/7)
\n" ); document.write( "step 3…we want to make 2nd.row.2nd.col.as
\n" ); document.write( "1..so..nr2=or2/14
\n" ); document.write( "1...... -1..... 5/7.... 9/7
\n" ); document.write( "0...... 1...... (-94/7)/14..... (-200/7)/14
\n" ); document.write( "0...... 7...... (-7-6*5/7)..... (-2-6*9/7)
\n" ); document.write( "step 4..we want to make 3rd.row.2nd.col.as
\n" ); document.write( "0…so….nr3=or3-7*r2
\n" ); document.write( "1...... -1..... (5/7).. (9/7)
\n" ); document.write( "0...... 1...... (-94/7)/14..... (-200/7)/14
\n" ); document.write( "0 0 (-79/7)-7*(-94/98) (68/7)-7*(-200/98)
\n" ); document.write( "step 5….we want to make 3rd.row.3rd.col.as
\n" ); document.write( "1…so….nr3=or3/(-32/7)
\n" ); document.write( "1...... -1..... (5/7).. (9/7)
\n" ); document.write( "0...... 1...... (-94/98)... (-200/98)
\n" ); document.write( "0...... 0..... 1...... -1
\n" ); document.write( "step 6…we want to make 1st/2nd.row 3rd.col.as
\n" ); document.write( "0..so..nr1=or1-5*r3/7...nr2=or2+94*r3/98
\n" ); document.write( "1...... -1..... 0..... 2
\n" ); document.write( "0...... 1...... 0...... -3
\n" ); document.write( "0...... 0...... 1...... -1
\n" ); document.write( "step7….we want to make 1st.row 2nd.col.as
\n" ); document.write( "0..so….nr1=or1+r2
\n" ); document.write( "1...... 0...... 0...... -1
\n" ); document.write( "0...... 1...... 0...... -3
\n" ); document.write( "0...... 0...... 1...... -1
\n" ); document.write( "so x=-1.....y=-3.....and z=-1...you can check back
\n" ); document.write( "YOU CAN SEE THE FOLLOWING ADDITIONAL MATERIAL FOR
\n" ); document.write( "REFERENCE
\n" ); document.write( "How do I perform the next required row operation on
\n" ); document.write( "the following matrix and provide only the next table:
\n" ); document.write( "x y z
\n" ); document.write( "1 28 14 245
\n" ); document.write( "0 3 7 42
\n" ); document.write( "0 7 7 -38
\n" ); document.write( "1 solutions
\n" ); document.write( "Answer 9892 by venugopalramana(370) About Me on
\n" ); document.write( "2005-11-25 08:01:32 (Show Source):
\n" ); document.write( "trust you want to solve the equations for x,y and z
\n" ); document.write( "and you are at this stage now....assuming that
\n" ); document.write( ".....our objective is to finally get the matrix if
\n" ); document.write( "possible into the following form ....(i am using
\n" ); document.write( "....to seperate the numbers with suitable gaps..your
\n" ); document.write( "typing is giving raise to uneven gaps bringing a
\n" ); document.write( "little lack of clarity)
\n" ); document.write( "1.....0.....0.....x
\n" ); document.write( "0.....1.....0.....y
\n" ); document.write( "0.....0.....1.....z
\n" ); document.write( "now we have
\n" ); document.write( "1......28.....14.....245
\n" ); document.write( "0.......3......7......42
\n" ); document.write( "0.......7......7.....-38
\n" ); document.write( "new row2=old row2/3.......to get 1 as required in
\n" ); document.write( "row2.so we get...
\n" ); document.write( "1......28.....14.....245
\n" ); document.write( "0......3/3....7/3....42/3
\n" ); document.write( "0.......7......7.....-38
\n" ); document.write( "new row3=oldrow3-7*row2 to get 0 as required in row3
\n" ); document.write( "1......28.....14...........245
\n" ); document.write( "0.......1.....7/3...........14
\n" ); document.write( "0......7-7*1..7-7*7/3......-38-7*14
\n" ); document.write( "new row3 = old row3/(-28/3)..to get 1 as required in
\n" ); document.write( "row3
\n" ); document.write( "1......28.....14...................245
\n" ); document.write( "0.......1.....7/3...................14
\n" ); document.write( "0.......0....(-28/3)/(-28/3).....(-136)/(-28/3)
\n" ); document.write( "this gives us finally in the following form
\n" ); document.write( "1......28.....14............245
\n" ); document.write( "0.......1.....7/3...........14
\n" ); document.write( "0.......0......1............102/7
\n" ); document.write( "now we go back in the same way to get 0 in row2 and
\n" ); document.write( "row3
\n" ); document.write( "new row2=old row2-row3*7/3...and new row1=old
\n" ); document.write( "row1-row3*14...so we get
\n" ); document.write( "1......28......14-1*14.......245-(102/7)*14
\n" ); document.write( "0.......1.......7/3-(7/3)*1...14-(102/7)*(7/3)
\n" ); document.write( "0.......0.........1.............102/7
\n" ); document.write( "the above on simplification gives us
\n" ); document.write( "1.......28.......0..........41
\n" ); document.write( "0........1.......0..........-20
\n" ); document.write( "0........0.......1..........102/7
\n" ); document.write( "now finally we take new row1=old row1-28*row2
\n" ); document.write( "1.......28-28*1......0.......41-(-28*20)
\n" ); document.write( "0........1.......0...........-20
\n" ); document.write( "0........0.......1...........102/7
\n" ); document.write( "so the final answer is
\n" ); document.write( "1......0.......0.......601
\n" ); document.write( "0......1.......0.......-20
\n" ); document.write( "0......0.......1.......102/7
\n" ); document.write( "which tells us that
\n" ); document.write( "1*x+0*y+0*z=x=601
\n" ); document.write( "0*x+1*y+0*z=y=-20
\n" ); document.write( "0*x+0*y+1*z=z=102/7
\n" ); document.write( "note that each and every transformation we did above
\n" ); document.write( "can be interpreted as given in the last statement
\n" ); document.write( "given above...this i hope will give you the insight of
\n" ); document.write( "the process at every step.you can also substitute
\n" ); document.write( "these values of x,y and z in each and every matrix
\n" ); document.write( "above to see that they satify all the equations given
\n" ); document.write( "by the different matrices..in general each mtrix can
\n" ); document.write( "be taken as a set of simltanous equations in x,y and
\n" ); document.write( "z...they can be written as follows..take column 1 is
\n" ); document.write( "for x,column 2 is for y and column 3 is for z.so the
\n" ); document.write( "first matrix you gave
\n" ); document.write( "1......28.....14.....245
\n" ); document.write( "0.......3......7......42
\n" ); document.write( "0.......7......7.....-38
\n" ); document.write( "tells us that
\n" ); document.write( "1*x+28*y+14*z=245....etc... \n" ); document.write( "