document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=64626>Question 64626</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/003002174x/algebrahomeworkh>Holt Algebra 2 w/Trig </A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> I have tried for 2 days to solve this one!  I have the answers, and they do solve the equations, so they are correct.  However, I am unable to produce the answers using matrix transformations. Any step-by-step help would be appreciated.\r<BR>\n" );
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document.write( "-2r + 6s -3t + 3v = -5<BR>\n" );
document.write( "4r - 3s + 2t -v = -3<BR>\n" );
document.write( "-3r - 4s - 3t + 4v = -15<BR>\n" );
document.write( "5r + 2s - 2t + 3v = -20\r<BR>\n" );
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document.write( "The solution is r,s,t,v = -2,1,3,-2 respectively. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #45207 by </B><A HREF=/tutors/aboutme.mpl?userid=venugopalramana><B>venugopalramana(3286)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=venugopalramana><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.geocities.com/venugopalramana/FREE_MATHS_ASSISTANCE.html valign=middle><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=45207\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> see the following and try.if you are still in difficulty please come back<BR>\n" );
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document.write( "Solve the system of linear equations by using the matrices: [9,3,4][x1]=7<BR>\n" );
document.write( "[4,3,4][x2]=8<BR>\n" );
document.write( "[1,1,1][x3]=3\r<BR>\n" );
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document.write( "9 3 4 7<BR>\n" );
document.write( "4 3 4 8<BR>\n" );
document.write( "1 1 1 3<BR>\n" );
document.write( "R1=R1/9<BR>\n" );
document.write( "1 0.333333333 0.444444444 0.777777778<BR>\n" );
document.write( "4 3 4 8<BR>\n" );
document.write( "1 1 1 3<BR>\n" );
document.write( "R2=R2-4R1…..R3=R3-R1<BR>\n" );
document.write( "1 0.333333333 0.444444444 0.777777778<BR>\n" );
document.write( "0 1.666666667 2.222222222 4.888888889<BR>\n" );
document.write( "0 0.666666667 0.555555556 2.222222222<BR>\n" );
document.write( "R2=R2/(5/3)<BR>\n" );
document.write( "1 0.333333333 0.444444444 0.777777778<BR>\n" );
document.write( "0 1 1.333333333 2.933333333<BR>\n" );
document.write( "0 0.666666667 0.555555556 2.222222222<BR>\n" );
document.write( "R3=R3-R2*2/3<BR>\n" );
document.write( "1 0.333333333 0.444444444 0.777777778<BR>\n" );
document.write( "0 1 1.333333333 2.933333333<BR>\n" );
document.write( "0 0 -0.333333333 0.266666667<BR>\n" );
document.write( "R3=-R3*3<BR>\n" );
document.write( "1 0.333333333 0.444444444 0.777777778<BR>\n" );
document.write( "0 1 1.333333333 2.933333333<BR>\n" );
document.write( "0 0 1 -0.8<BR>\n" );
document.write( "R2=R2-R3*4/3….R1=R1-R3*4/9<BR>\n" );
document.write( "1 0.333333333 0 1.133333333<BR>\n" );
document.write( "0 1 0 4<BR>\n" );
document.write( "0 0 1 -0.8<BR>\n" );
document.write( "R1=R1-R2/3<BR>\n" );
document.write( "1 0 0 -0.2<BR>\n" );
document.write( "0 1 0 4<BR>\n" );
document.write( "0 0 1 -0.8<BR>\n" );
document.write( "HENCE X1=-0.2 X2=4 X3=-0.8\r<BR>\n" );
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document.write( "Linear_Algebra/35842: Use the Gauss Jordan method to<BR>\n" );
document.write( "solve the system of equations:<BR>\n" );
document.write( "x+y+2z=7<BR>\n" );
document.write( "3x-y+z=10<BR>\n" );
document.write( "2x+y-3z=-6<BR>\n" );
document.write( "1 solutions<BR>\n" );
document.write( "Answer 21966 by venugopalramana(2734) About Me on<BR>\n" );
document.write( "2006-05-02 06:35:47 (Show Source):<BR>\n" );
document.write( "You can put this solution on YOUR website!<BR>\n" );
document.write( "Use the Gauss Jordan method to solve the system of<BR>\n" );
document.write( "equations:<BR>\n" );
document.write( "x+y+2z=7.......... I<BR>\n" );
document.write( "3x-y+z=10............II<BR>\n" );
document.write( "2x+y-3z=-6................III<BR>\n" );
document.write( "THE COEFFICIENT AND AUGMENTED MATRIX IS<BR>\n" );
document.write( "X,Y,Z,CONSTANT<BR>\n" );
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document.write( "1,1,2,7...............MEANS EQN.I<BR>\n" );
document.write( "3,-1,1,10.............MEANS EQN.II<BR>\n" );
document.write( "2,1,-3,-6.............MEANS EQN.III<BR>\n" );
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document.write( "SEE THE FOLLOWING EXAMPLE SOLVED FOR A PROBLEM SIMILAR<BR>\n" );
document.write( "TO YOURS WHICH IS<BR>\n" );
document.write( "ESSENTIALLY SAME BUT FOR THE NUMBERS , FOLLOW STEP BY<BR>\n" );
document.write( "STEP AND DO BY YOUR SELF,THERE BY YOU WILL LEARN HOW<BR>\n" );
document.write( "TO DO AND BECOME AN EXPERT BY YOUR SELF!!!!IF YOU DO<BR>\n" );
document.write( "NOT UNDERSTAND ASK WHERE IT IS NOT CLEAR...<BR>\n" );
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document.write( "Linear_Algebra/20750: Hi, I'm in homeschooling and I'm<BR>\n" );
document.write( "having trouble with matrices. I was wondering how to<BR>\n" );
document.write( "solve the problem where you have to find the x,y, and<BR>\n" );
document.write( "z values in the matrix:<BR>\n" );
document.write( "[7 -7 5 | 9]<BR>\n" );
document.write( "[9 5 -7 | -17]<BR>\n" );
document.write( "[6 1 -7 | -2]<BR>\n" );
document.write( "I'd appreciate the help. Thank you!<BR>\n" );
document.write( "Caitlyn Reese<BR>\n" );
document.write( "1 solutions<BR>\n" );
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document.write( "Answer 9969 by venugopalramana(390) on 2005-11-28<BR>\n" );
document.write( "07:14:11 (Show Source):<BR>\n" );
document.write( "the 4 column heads represent x,y,z and constant term<BR>\n" );
document.write( "in the matrix of system of eqns.<BR>\n" );
document.write( "then each row gives us one eqn.like say row 1 gives us<BR>\n" );
document.write( "that 7x-7y+5z=9..etc…<BR>\n" );
document.write( "hence if we can make the matrix to become<BR>\n" );
document.write( "1 0 0 ?<BR>\n" );
document.write( "0 1 0 ??<BR>\n" );
document.write( "0 0 1 ???<BR>\n" );
document.write( "then from the explantion given above it means<BR>\n" );
document.write( "1x=?.1y=?? And 1z=???<BR>\n" );
document.write( "so we try to transform the matrix in to that form..by<BR>\n" );
document.write( "the following steps.<BR>\n" );
document.write( "in fact using the above explanation,you can see that<BR>\n" );
document.write( "what we do at each step is just<BR>\n" );
document.write( "divide each eqn. with a constant/add/subtract etc<BR>\n" );
document.write( "which does not change the basic<BR>\n" );
document.write( "eqn.for ex. dividing row 1 by 7 means change the given<BR>\n" );
document.write( "eqn.7x-7y+5z=9 to x-y+5z/7=9/7<BR>\n" );
document.write( "legend:- or1 means old row 1..nr1 means new row 1…r1<BR>\n" );
document.write( "means the existing row 1 please note that no changes<BR>\n" );
document.write( "are made in rows other than those mentioned at each<BR>\n" );
document.write( "step.<BR>\n" );
document.write( "start with given matrix …<BR>\n" );
document.write( "7...... -7..... 5...... 9<BR>\n" );
document.write( "9...... 5...... -7..... -17<BR>\n" );
document.write( "6...... 1...... -7..... -2<BR>\n" );
document.write( "step 1…we want to make 1st.row 1st.column as<BR>\n" );
document.write( "1….so….nr1=or1/7...<BR>\n" );
document.write( "1...... -1..... (5/7).. (9/7)<BR>\n" );
document.write( "9...... 5...... -7..... -17<BR>\n" );
document.write( "6...... 1...... -7..... -2<BR>\n" );
document.write( "step 2..we want to make 2nd/3rd.rows,col.1 as<BR>\n" );
document.write( "0...so...nr2=or2-9*r1........nr3=or3-6*r1<BR>\n" );
document.write( "1...... -1..... (5/7).. (9/7)<BR>\n" );
document.write( "0... 14..... (-7-9*5/7).... (-17-9*9/7)<BR>\n" );
document.write( "0...... 7...... (-7-6*5/7)..... (-2-6*9/7)<BR>\n" );
document.write( "step 3…we want to make 2nd.row.2nd.col.as<BR>\n" );
document.write( "1..so..nr2=or2/14<BR>\n" );
document.write( "1...... -1..... 5/7.... 9/7<BR>\n" );
document.write( "0...... 1...... (-94/7)/14..... (-200/7)/14<BR>\n" );
document.write( "0...... 7...... (-7-6*5/7)..... (-2-6*9/7)<BR>\n" );
document.write( "step 4..we want to make 3rd.row.2nd.col.as<BR>\n" );
document.write( "0…so….nr3=or3-7*r2<BR>\n" );
document.write( "1...... -1..... (5/7).. (9/7)<BR>\n" );
document.write( "0...... 1...... (-94/7)/14..... (-200/7)/14<BR>\n" );
document.write( "0 0 (-79/7)-7*(-94/98) (68/7)-7*(-200/98)<BR>\n" );
document.write( "step 5….we want to make 3rd.row.3rd.col.as<BR>\n" );
document.write( "1…so….nr3=or3/(-32/7)<BR>\n" );
document.write( "1...... -1..... (5/7).. (9/7)<BR>\n" );
document.write( "0...... 1...... (-94/98)... (-200/98)<BR>\n" );
document.write( "0...... 0..... 1...... -1<BR>\n" );
document.write( "step 6…we want to make 1st/2nd.row 3rd.col.as<BR>\n" );
document.write( "0..so..nr1=or1-5*r3/7...nr2=or2+94*r3/98<BR>\n" );
document.write( "1...... -1..... 0..... 2<BR>\n" );
document.write( "0...... 1...... 0...... -3<BR>\n" );
document.write( "0...... 0...... 1...... -1<BR>\n" );
document.write( "step7….we want to make 1st.row 2nd.col.as<BR>\n" );
document.write( "0..so….nr1=or1+r2<BR>\n" );
document.write( "1...... 0...... 0...... -1<BR>\n" );
document.write( "0...... 1...... 0...... -3<BR>\n" );
document.write( "0...... 0...... 1...... -1<BR>\n" );
document.write( "so x=-1.....y=-3.....and z=-1...you can check back<BR>\n" );
document.write( "YOU CAN SEE THE FOLLOWING ADDITIONAL MATERIAL FOR<BR>\n" );
document.write( "REFERENCE<BR>\n" );
document.write( "How do I perform the next required row operation on<BR>\n" );
document.write( "the following matrix and provide only the next table:<BR>\n" );
document.write( "x y z<BR>\n" );
document.write( "1 28 14 245<BR>\n" );
document.write( "0 3 7 42<BR>\n" );
document.write( "0 7 7 -38<BR>\n" );
document.write( "1 solutions<BR>\n" );
document.write( "Answer 9892 by venugopalramana(370) About Me on<BR>\n" );
document.write( "2005-11-25 08:01:32 (Show Source):<BR>\n" );
document.write( "trust you want to solve the equations for x,y and z<BR>\n" );
document.write( "and you are at this stage now....assuming that<BR>\n" );
document.write( ".....our objective is to finally get the matrix if<BR>\n" );
document.write( "possible into the following form ....(i am using<BR>\n" );
document.write( "....to seperate the numbers with suitable gaps..your<BR>\n" );
document.write( "typing is giving raise to uneven gaps bringing a<BR>\n" );
document.write( "little lack of clarity)<BR>\n" );
document.write( "1.....0.....0.....x<BR>\n" );
document.write( "0.....1.....0.....y<BR>\n" );
document.write( "0.....0.....1.....z<BR>\n" );
document.write( "now we have<BR>\n" );
document.write( "1......28.....14.....245<BR>\n" );
document.write( "0.......3......7......42<BR>\n" );
document.write( "0.......7......7.....-38<BR>\n" );
document.write( "new row2=old row2/3.......to get 1 as required in<BR>\n" );
document.write( "row2.so we get...<BR>\n" );
document.write( "1......28.....14.....245<BR>\n" );
document.write( "0......3/3....7/3....42/3<BR>\n" );
document.write( "0.......7......7.....-38<BR>\n" );
document.write( "new row3=oldrow3-7*row2 to get 0 as required in row3<BR>\n" );
document.write( "1......28.....14...........245<BR>\n" );
document.write( "0.......1.....7/3...........14<BR>\n" );
document.write( "0......7-7*1..7-7*7/3......-38-7*14<BR>\n" );
document.write( "new row3 = old row3/(-28/3)..to get 1 as required in<BR>\n" );
document.write( "row3<BR>\n" );
document.write( "1......28.....14...................245<BR>\n" );
document.write( "0.......1.....7/3...................14<BR>\n" );
document.write( "0.......0....(-28/3)/(-28/3).....(-136)/(-28/3)<BR>\n" );
document.write( "this gives us finally in the following form<BR>\n" );
document.write( "1......28.....14............245<BR>\n" );
document.write( "0.......1.....7/3...........14<BR>\n" );
document.write( "0.......0......1............102/7<BR>\n" );
document.write( "now we go back in the same way to get 0 in row2 and<BR>\n" );
document.write( "row3<BR>\n" );
document.write( "new row2=old row2-row3*7/3...and new row1=old<BR>\n" );
document.write( "row1-row3*14...so we get<BR>\n" );
document.write( "1......28......14-1*14.......245-(102/7)*14<BR>\n" );
document.write( "0.......1.......7/3-(7/3)*1...14-(102/7)*(7/3)<BR>\n" );
document.write( "0.......0.........1.............102/7<BR>\n" );
document.write( "the above on simplification gives us<BR>\n" );
document.write( "1.......28.......0..........41<BR>\n" );
document.write( "0........1.......0..........-20<BR>\n" );
document.write( "0........0.......1..........102/7<BR>\n" );
document.write( "now finally we take new row1=old row1-28*row2<BR>\n" );
document.write( "1.......28-28*1......0.......41-(-28*20)<BR>\n" );
document.write( "0........1.......0...........-20<BR>\n" );
document.write( "0........0.......1...........102/7<BR>\n" );
document.write( "so the final answer is<BR>\n" );
document.write( "1......0.......0.......601<BR>\n" );
document.write( "0......1.......0.......-20<BR>\n" );
document.write( "0......0.......1.......102/7<BR>\n" );
document.write( "which tells us that<BR>\n" );
document.write( "1*x+0*y+0*z=x=601<BR>\n" );
document.write( "0*x+1*y+0*z=y=-20<BR>\n" );
document.write( "0*x+0*y+1*z=z=102/7<BR>\n" );
document.write( "note that each and every transformation we did above<BR>\n" );
document.write( "can be interpreted as given in the last statement<BR>\n" );
document.write( "given above...this i hope will give you the insight of<BR>\n" );
document.write( "the process at every step.you can also substitute<BR>\n" );
document.write( "these values of x,y and z in each and every matrix<BR>\n" );
document.write( "above to see that they satify all the equations given<BR>\n" );
document.write( "by the different matrices..in general each mtrix can<BR>\n" );
document.write( "be taken as a set of simltanous equations in x,y and<BR>\n" );
document.write( "z...they can be written as follows..take column 1 is<BR>\n" );
document.write( "for x,column 2 is for y and column 3 is for z.so the<BR>\n" );
document.write( "first matrix you gave<BR>\n" );
document.write( "1......28.....14.....245<BR>\n" );
document.write( "0.......3......7......42<BR>\n" );
document.write( "0.......7......7.....-38<BR>\n" );
document.write( "tells us that<BR>\n" );
document.write( "1*x+28*y+14*z=245....etc...<BR>\n" );
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