document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=918070>Question 918070</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A security firm borrowed $900,000 for business expansion. Some of the money was borrowed at 5%, some at 6%,and some at 7% simple annual interest.The total annual interest is $52,000 and the amount borrowed at 6% is one and a half of that borrowed at 7%.<BR>\n" );
document.write( "a. Create a system of equations which captures the essence of the information given. <BR>\n" );
document.write( "b. Write the system is matrix form? <BR>\n" );
document.write( "c. Using the inverse method,find the amount borrowed at each rate.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #556887 by </B><A HREF=/tutors/aboutme.mpl?userid=richwmiller><B>richwmiller(17219)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richwmiller><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=richwmiller><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=556887\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> x+y+z=900000,<BR>\n" );
document.write( "0.06*x+0.05*y+0.07*z=52000,<BR>\n" );
document.write( "x=1.5z\r<BR>\n" );
document.write( "\n" );
document.write( "1,1,1,900000<BR>\n" );
document.write( ".06,.05,.07,52000<BR>\n" );
document.write( "1,0,-1.5,0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "900000 52000 0<BR>\n" );
document.write( "1 1 1<BR>\n" );
document.write( "0.06 0.05 0.07<BR>\n" );
document.write( "1 0 -1.5<BR>\n" );
document.write( "1st=0.05 0.07 0 -1.5<BR>\n" );
document.write( "2nd=0.06 0.07 1 -1.5<BR>\n" );
document.write( "3rd=0.06 0.05 1 0<BR>\n" );
document.write( "4th=1 1 0 -1.5<BR>\n" );
document.write( "5th=1 1 1 -1.5<BR>\n" );
document.write( "6th=1 1 1 0<BR>\n" );
document.write( "7th=1 1 0.05 0.07<BR>\n" );
document.write( "8th=1 1 0.06 0.07<BR>\n" );
document.write( "9th=1 1 0.06 0.05<BR>\n" );
document.write( "all minors<BR>\n" );
document.write( "1=0.05 0.07 0 -1.5<BR>\n" );
document.write( "2=0.06 0.07 1 -1.5<BR>\n" );
document.write( "3=0.06 0.05 1 0<BR>\n" );
document.write( "4=1 1 0 -1.5<BR>\n" );
document.write( "5=1 1 1 -1.5<BR>\n" );
document.write( "6=1 1 1 0<BR>\n" );
document.write( "7=1 1 0.05 0.07<BR>\n" );
document.write( "8=1 1 0.06 0.07<BR>\n" );
document.write( "9=1 1 0.06 0.05<BR>\n" );
document.write( "mult ad<BR>\n" );
document.write( " ad-bc  1*4-2*3<BR>\n" );
document.write( "1--<BR>\n" );
document.write( "0.05 0.07 0 -1.5<BR>\n" );
document.write( "(0.05*-1.5)-(0.07*0)<BR>\n" );
document.write( "-0.075-(0.0) =-0.075<BR>\n" );
document.write( "2--<BR>\n" );
document.write( "0.06 0.07 1 -1.5<BR>\n" );
document.write( "(0.06*-1.5)-(0.07*1)<BR>\n" );
document.write( "-0.09-(0.07) =-0.16<BR>\n" );
document.write( "3--<BR>\n" );
document.write( "0.06 0.05 1 0<BR>\n" );
document.write( "(0.06*0)-(0.05*1)<BR>\n" );
document.write( "0.0-(0.05) =-0.05<BR>\n" );
document.write( "4--<BR>\n" );
document.write( "1 1 0 -1.5<BR>\n" );
document.write( "(1*-1.5)-(1*0)<BR>\n" );
document.write( "-1.5-(0) =-1.5<BR>\n" );
document.write( "5--<BR>\n" );
document.write( "1 1 1 -1.5<BR>\n" );
document.write( "(1*-1.5)-(1*1)<BR>\n" );
document.write( "-1.5-(1) =-2.5<BR>\n" );
document.write( "6--<BR>\n" );
document.write( "1 1 1 0<BR>\n" );
document.write( "(1*0)-(1*1)<BR>\n" );
document.write( "0-(1) =-1<BR>\n" );
document.write( "7--<BR>\n" );
document.write( "1 1 0.05 0.07<BR>\n" );
document.write( "(1*0.07)-(1*0.05)<BR>\n" );
document.write( "0.07-(0.05) =0.02<BR>\n" );
document.write( "8--<BR>\n" );
document.write( "1 1 0.06 0.07<BR>\n" );
document.write( "(1*0.07)-(1*0.06)<BR>\n" );
document.write( "0.07-(0.06) =0.01<BR>\n" );
document.write( "9--<BR>\n" );
document.write( "1 1 0.06 0.05<BR>\n" );
document.write( "(1*0.05)-(1*0.06)<BR>\n" );
document.write( "0.05-(0.06) =-0.01<BR>\n" );
document.write( " matrix of minors<BR>\n" );
document.write( "-0.075 -0.16 -0.05<BR>\n" );
document.write( "-1.5 -2.5 -1<BR>\n" );
document.write( "0.02 0.01 -0.01<BR>\n" );
document.write( "original<BR>\n" );
document.write( "1 1 1<BR>\n" );
document.write( "0.06 0.05 0.07<BR>\n" );
document.write( "1 0 -1.5<BR>\n" );
document.write( "matrix of cofactors-switching signs<BR>\n" );
document.write( "-0.075 same -0.075<BR>\n" );
document.write( "-0.16 switch 0.16<BR>\n" );
document.write( "-0.05 same -0.05<BR>\n" );
document.write( "-1.5 switch 1.5<BR>\n" );
document.write( "-2.5 same -2.5<BR>\n" );
document.write( "-1 switch 1<BR>\n" );
document.write( "0.02 same 0.02<BR>\n" );
document.write( "0.01 switch -0.01<BR>\n" );
document.write( "-0.01 same -0.01<BR>\n" );
document.write( "matrix of cofactors<BR>\n" );
document.write( "-0.075 0.16 -0.05<BR>\n" );
document.write( "1.5 -2.5 1<BR>\n" );
document.write( "0.02 -0.01 -0.01<BR>\n" );
document.write( "adjoint transpose<BR>\n" );
document.write( "-0.075 1.5 0.02<BR>\n" );
document.write( "0.16 -2.5 -0.01<BR>\n" );
document.write( "-0.05 1 -0.01<BR>\n" );
document.write( "original by row<BR>\n" );
document.write( "(row 1)1 1 1<BR>\n" );
document.write( "(row 2)0.06 0.05 0.07<BR>\n" );
document.write( "(row 3)1 0 -1.5<BR>\n" );
document.write( "original by column<BR>\n" );
document.write( "(column 1)1 0.06 1<BR>\n" );
document.write( "(column 2)1 0.05 0<BR>\n" );
document.write( "(column 3)1 0.07 -1.5\r<BR>\n" );
document.write( "\n" );
document.write( "1) determinant using  row three<BR>\n" );
document.write( "1*0.02=0.02<BR>\n" );
document.write( "0*-0.01=0<BR>\n" );
document.write( "-1.5*-0.01=0.015<BR>\n" );
document.write( "determinant=0.035<BR>\n" );
document.write( "and finally the inverse<BR>\n" );
document.write( "-0.075/0.035 1.5/0.035 0.02/0.035<BR>\n" );
document.write( "0.16/0.035 -2.5/0.035 -0.01/0.035<BR>\n" );
document.write( "-0.05/0.035 1/0.035 -0.01/0.035\r<BR>\n" );
document.write( "\n" );
document.write( "2) same determinant using column 2<BR>\n" );
document.write( "1*0.16=0.16<BR>\n" );
document.write( "0.05*-2.5=-0.125<BR>\n" );
document.write( "0*-0.01=0<BR>\n" );
document.write( "determinant=0.035<BR>\n" );
document.write( " the inverse in fractions<BR>\n" );
document.write( "-0.075/0.035 1.5/0.035 0.02/0.035<BR>\n" );
document.write( "0.16/0.035 -2.5/0.035 -0.01/0.035<BR>\n" );
document.write( "-0.05/0.035 1/0.035 -0.01/0.035<BR>\n" );
document.write( " the inverse in decimals<BR>\n" );
document.write( "-2.14285714 42.8571429 0.57142857<BR>\n" );
document.write( "4.57142857 -71.4285714 -0.28571429<BR>\n" );
document.write( "-1.42857143 28.5714286 -0.28571429<BR>\n" );
document.write( "900000 52000 0<BR>\n" );
document.write( "column one<BR>\n" );
document.write( "-2.14285714 42.8571429 0.57142857<BR>\n" );
document.write( "-1928571.43 2228571.43 0.0<BR>\n" );
document.write( "solutions<BR>\n" );
document.write( "x=300000.0 y= 400000 z= 200000\r<BR>\n" );
document.write( "\n" );
document.write( "x = $300000 at 6%<BR>\n" );
document.write( "y = $400000 at 5%<BR>\n" );
document.write( "z = $200000 at 7%\r<BR>\n" );
document.write( "\n" );
document.write( "Here is the gauss jordan method<BR>\n" );
document.write( "1,1,1,900000<BR>\n" );
document.write( ".06,.05,.07,52000<BR>\n" );
document.write( "1,0,-1.5,0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "add  down (-0.06) *row 1 to row 2<BR>\n" );
document.write( "1,1,1,900000<BR>\n" );
document.write( "0,-0.01/1,0.01/1,-2000<BR>\n" );
document.write( "1,0,-1.5/1,0\r<BR>\n" );
document.write( "\n" );
document.write( "add  down (-1) *row 1 to row 3<BR>\n" );
document.write( "1,1,1,900000<BR>\n" );
document.write( "0,-0.01/1,0.01/1,-2000<BR>\n" );
document.write( "0,-1,-2.5/1,-900000\r<BR>\n" );
document.write( "\n" );
document.write( "divide row 2 by -0.01<BR>\n" );
document.write( "1,1,1,900000<BR>\n" );
document.write( "0,1,0.01/-0.01,-2000/-0.01<BR>\n" );
document.write( "0,-1,-2.5/1,-900000\r<BR>\n" );
document.write( "\n" );
document.write( "add  down (1) *row 2 to row 3<BR>\n" );
document.write( "1,1,1,900000<BR>\n" );
document.write( "0,1,0.01/-0.01,2000/0.01<BR>\n" );
document.write( "0,0,0.035/-0.01,-700000\r<BR>\n" );
document.write( "\n" );
document.write( "divide row 3 by 0.035/-0.01<BR>\n" );
document.write( "1,1,1,900000<BR>\n" );
document.write( "0,1,0.01/-0.01,2000/0.01<BR>\n" );
document.write( "0,0,1,7000.0/0.035\r<BR>\n" );
document.write( "\n" );
document.write( "We now have the value for the last variable.<BR>\n" );
document.write( "We will work our way up and get the other solutions.\r<BR>\n" );
document.write( "\n" );
document.write( "add up  (-0.01/-0.01) *row 3 to row 2<BR>\n" );
document.write( "1,1,1,900000<BR>\n" );
document.write( "0,1,0,-1.4/-0.0000035<BR>\n" );
document.write( "0,0,1,7000.0/0.035\r<BR>\n" );
document.write( "\n" );
document.write( "add up  (-1) *row 3 to row 1<BR>\n" );
document.write( "1,1,0,700000<BR>\n" );
document.write( "0,1,0,1.4/0.0000035<BR>\n" );
document.write( "0,0,1,7000.0/0.035\r<BR>\n" );
document.write( "\n" );
document.write( "add up  (-1) *row 2 to row 1<BR>\n" );
document.write( "1,0,0,1.05/0.0000035<BR>\n" );
document.write( "0,1,0,1.4/0.0000035<BR>\n" );
document.write( "0,0,1,7000.0/0.035\r<BR>\n" );
document.write( "\n" );
document.write( "final<BR>\n" );
document.write( "1,0,0,1.05/0.0000035<BR>\n" );
document.write( "0,1,0,1.4/0.0000035<BR>\n" );
document.write( "0,0,1,7000.0/0.035\r<BR>\n" );
document.write( "\n" );
document.write( "1,0,0,1.05/0.0000035 = 300000.0<BR>\n" );
document.write( "0,1,0,1.4/0.0000035 = 400000.0<BR>\n" );
document.write( "0,0,1,7000.0/0.035 = 200000.0<BR>\n" );
document.write( "\"1.05/0.0000035\",\"1.4/0.0000035\",\"7000.0/0.035\"<BR>\n" );
document.write( "(1.05/0.0000035,1.4/0.0000035,7000.0/0.035)<BR>\n" );
document.write( "(300000.0,400000.0,200000.0)\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );