{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis " -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" 258 267 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 268 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 15 "Cramer's Rule " }}{PARA 0 "" 0 " " {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 " " {TEXT -1 19 "Version: 26.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 80 "Cr amer's rule for the solution of a system of 2 linear equations in 2 va riables " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 57 "Consider the problem of solving the system of equati ons: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([a [1]*x+b[1]*y = c[1], ``],[a[2]*x+b[2]*y = c[2], ``]);" "6#-%*PIECEWISE G6$7$/,&*&&%\"aG6#\"\"\"F-%\"xGF-F-*&&%\"bG6#F-F-%\"yGF-F-&%\"cG6#F-%! G7$/,&*&&F+6#\"\"#F-F.F-F-*&&F16#F>F-F3F-F-&F56#F>F7" }{TEXT -1 14 " - ------ (i). " }}{PARA 0 "" 0 "" {TEXT -1 51 "This system can be repres ented in the matrix form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[a[1], b[1]], [a[2], b[2]]])*matrix([[x], [y]]) = matrix([[c[1]], [c[2]]]);" "6#/*&-%'matrixG6#7$7$&%\"aG6#\"\"\"&%\" bG6#F-7$&F+6#\"\"#&F/6#F4F--F&6#7$7#%\"xG7#%\"yGF--F&6#7$7#&%\"cG6#F-7 #&FC6#F4" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "The correspon ding augmented matrix is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[a[1], b[1], `|`, c[1]], [a[2], b[2], `|`, c[2] ]]);" "6#-%'matrixG6#7$7&&%\"aG6#\"\"\"&%\"bG6#F+%\"|grG&%\"cG6#F+7&&F )6#\"\"#&F-6#F6F/&F16#F6" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We now perform Gaussian eliminatio n with the given row operations. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([R[1] = R[ 1]*`.`*a[2], ``],[R[2] = R[2]*`.`*a[1], ``]);" "6#-%*PIECEWISEG6$7$/&% \"RG6#\"\"\"*(&F)6#F+F+%\".GF+&%\"aG6#\"\"#F+%!G7$/&F)6#F3*(&F)6#F3F+F /F+&F16#F+F+F4" }{TEXT -1 4 "... " }{XPPEDIT 18 0 "matrix([[a[1]*a[2], a[2]*b[1], `|`, a[2]*c[1]], [a[1]*a[2], a[1]*b[2], `|`, a[1]*c[2]]]); " "6#-%'matrixG6#7$7&*&&%\"aG6#\"\"\"F,&F*6#\"\"#F,*&&F*6#F/F,&%\"bG6# F,F,%\"|grG*&&F*6#F/F,&%\"cG6#F,F,7&*&&F*6#F,F,&F*6#F/F,*&&F*6#F,F,&F4 6#F/F,F6*&&F*6#F,F,&F;6#F/F," }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[2]= R[2]-R[1]" "6#/&%\"RG6#\"\"#,&&F%6#F'\"\"\"&F%6#F+!\"\"" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "matrix([[a[1]*a[2], a[2]*b[1], `|`, a[2]*c[1] ], [0, a[1]*b[2]-a[2]*b[1], `|`, a[1]*c[2]-a[2]*c[1]]]);" "6#-%'matrix G6#7$7&*&&%\"aG6#\"\"\"F,&F*6#\"\"#F,*&&F*6#F/F,&%\"bG6#F,F,%\"|grG*&& F*6#F/F,&%\"cG6#F,F,7&\"\"!,&*&&F*6#F,F,&F46#F/F,F,*&&F*6#F/F,&F46#F,F ,!\"\"F6,&*&&F*6#F,F,&F;6#F/F,F,*&&F*6#F/F,&F;6#F,F,FJ" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "The \+ second row corresponds to the equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(a[1]*b[2]-a[2]*b[1])*y = a[1]*c[2]-a[2]*c[1];" "6#/*&,&*&&%\"aG6#\"\"\"F*&%\"bG6#\"\"#F*F**&&F(6#F.F*&F,6#F*F*!\"\"F* %\"yGF*,&*&&F(6#F*F*&%\"cG6#F.F*F**&&F(6#F.F*&F;6#F*F*F4" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 13 "which gives: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "y = (a[1]*c[2]-a[2]*c[1])/(a[1]*b[2]- a[2]*b[1]);" "6#/%\"yG*&,&*&&%\"aG6#\"\"\"F+&%\"cG6#\"\"#F+F+*&&F)6#F/ F+&F-6#F+F+!\"\"F+,&*&&F)6#F+F+&%\"bG6#F/F+F+*&&F)6#F/F+&F;6#F+F+F5F5 " }{XPPEDIT 18 0 "`` = det*matrix([[a[1], c[1]], [a[2], c[2]]])/(det*m atrix([[a[1], b[1]], [a[2], b[2]]]));" "6#/%!G*(%$detG\"\"\"-%'matrixG 6#7$7$&%\"aG6#F'&%\"cG6#F'7$&F.6#\"\"#&F16#F6F'*&F&F'-F)6#7$7$&F.6#F'& %\"bG6#F'7$&F.6#F6&FA6#F6F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "Notice that the determin ant in the numerator can be obtained from the determinant in the denom inator by replacing the column " }{XPPEDIT 18 0 "matrix([[b[1]],[b[2]] ])" "6#-%'matrixG6#7$7#&%\"bG6#\"\"\"7#&F)6#\"\"#" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "matrix([[c[1]], [c[2]]]);" "6#-%'matrixG6#7$7#&%\"cG6# \"\"\"7#&F)6#\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 56 "The system of equations (i) can be writt en in the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "P IECEWISE([b[2]*y+a[2]*x = c[2], ``],[b[1]*y+a[1]*x = c[1], ``]);" "6#- %*PIECEWISEG6$7$/,&*&&%\"bG6#\"\"#\"\"\"%\"yGF.F.*&&%\"aG6#F-F.%\"xGF. F.&%\"cG6#F-%!G7$/,&*&&F+6#F.F.F/F.F.*&&F26#F.F.F4F.F.&F66#F.F8" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "The corresponding matrix equation for the system in this \+ new form is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matr ix([[b[2], a[2]], [b[1], a[1]]])*matrix([[y], [x]]) = matrix([[c[2]], \+ [c[1]]]);" "6#/*&-%'matrixG6#7$7$&%\"bG6#\"\"#&%\"aG6#F-7$&F+6#\"\"\"& F/6#F4F4-F&6#7$7#%\"yG7#%\"xGF4-F&6#7$7#&%\"cG6#F-7#&FC6#F4" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "The corresponding augmented ma trix is: " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "matrix([[b[2], a[2], `|` , c[2]], [b[1], a[1], `|`, c[1]]]);" "6#-%'matrixG6#7$7&&%\"bG6#\"\"#& %\"aG6#F+%\"|grG&%\"cG6#F+7&&F)6#\"\"\"&F-6#F6F/&F16#F6" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We \+ now perform Gaussian elimination with the given row operations. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "PIECEWISE([R[1] = R[1]*`.`*b[1], ``],[R[2] = R[2]*`.`*b [2], ``]);" "6#-%*PIECEWISEG6$7$/&%\"RG6#\"\"\"*(&F)6#F+F+%\".GF+&%\"b G6#F+F+%!G7$/&F)6#\"\"#*(&F)6#F8F+F/F+&F16#F8F+F3" }{TEXT -1 4 "... " }{XPPEDIT 18 0 "matrix([[b[1]*b[2], a[2]*b[1], `|`, b[1]*c[2]], [b[1]* b[2], a[1]*b[2], `|`, b[2]*c[1]]]);" "6#-%'matrixG6#7$7&*&&%\"bG6#\"\" \"F,&F*6#\"\"#F,*&&%\"aG6#F/F,&F*6#F,F,%\"|grG*&&F*6#F,F,&%\"cG6#F/F,7 &*&&F*6#F,F,&F*6#F/F,*&&F26#F,F,&F*6#F/F,F6*&&F*6#F/F,&F;6#F,F," } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[2]=R[2]-R[1]" "6#/&%\"RG6#\"\"#,&&F%6 #F'\"\"\"&F%6#F+!\"\"" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "matrix([[ b[1]*b[2], a[2]*b[1], `|`, b[1]*c[2]], [0, a[1]*b[2]-a[2]*b[1], `|`, b [2]*c[1]-b[1]*c[2]]]);" "6#-%'matrixG6#7$7&*&&%\"bG6#\"\"\"F,&F*6#\"\" #F,*&&%\"aG6#F/F,&F*6#F,F,%\"|grG*&&F*6#F,F,&%\"cG6#F/F,7&\"\"!,&*&&F2 6#F,F,&F*6#F/F,F,*&&F26#F/F,&F*6#F,F,!\"\"F6,&*&&F*6#F/F,&F;6#F,F,F,*& &F*6#F,F,&F;6#F/F,FJ" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 43 "The second row corresponds to the equ ation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(a[1]*b[2]- a[2]*b[1])*x = b[2]*c[1]-b[1]*c[2];" "6#/*&,&*&&%\"aG6#\"\"\"F*&%\"bG6 #\"\"#F*F**&&F(6#F.F*&F,6#F*F*!\"\"F*%\"xGF*,&*&&F,6#F.F*&%\"cG6#F*F*F **&&F,6#F*F*&F;6#F.F*F4" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 13 "which gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " x = (b[2]*c[1]-b[1]*c[2])/(a[1]*b[2]-a[2]*b[1]);" "6#/%\"xG*&,&*&&%\"b G6#\"\"#\"\"\"&%\"cG6#F,F,F,*&&F)6#F,F,&F.6#F+F,!\"\"F,,&*&&%\"aG6#F,F ,&F)6#F+F,F,*&&F96#F+F,&F)6#F,F,F5F5" }{XPPEDIT 18 0 "`` = det*matrix( [[c[1], b[1]], [c[2], b[2]]])/(det*matrix([[a[1], b[1]], [a[2], b[2]]] ));" "6#/%!G*(%$detG\"\"\"-%'matrixG6#7$7$&%\"cG6#F'&%\"bG6#F'7$&F.6# \"\"#&F16#F6F'*&F&F'-F)6#7$7$&%\"aG6#F'&F16#F'7$&F?6#F6&F16#F6F'!\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 125 "Notice that the det erminant in the numerator can be obtained from the determinant in the \+ denominator by replacing the column " }{XPPEDIT 18 0 "matrix([[a[1]], \+ [a[2]]]);" "6#-%'matrixG6#7$7#&%\"aG6#\"\"\"7#&F)6#\"\"#" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "matrix([[c[1]], [c[2]]]);" "6#-%'matrixG6#7$7#& %\"cG6#\"\"\"7#&F)6#\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Summary " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 25 "The system of equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([a[1]*x+b[1]*y = c[1], ``],[a[2]*x+b[2 ]*y = c[2], ``]);" "6#-%*PIECEWISEG6$7$/,&*&&%\"aG6#\"\"\"F-%\"xGF-F-* &&%\"bG6#F-F-%\"yGF-F-&%\"cG6#F-%!G7$/,&*&&F+6#\"\"#F-F.F-F-*&&F16#F>F -F3F-F-&F56#F>F7" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 27 "whic h has the matrix form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[a[1], b[1]], [a[2], b[2]]])*matrix([[x], [y]]) = matrix ([[c[1]], [c[2]]]);" "6#/*&-%'matrixG6#7$7$&%\"aG6#\"\"\"&%\"bG6#F-7$& F+6#\"\"#&F/6#F4F--F&6#7$7#%\"xG7#%\"yGF--F&6#7$7#&%\"cG6#F-7#&FC6#F4 " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 19 "has the solutions: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = D[x]/D;" "6#/ %\"xG*&&%\"DG6#F$\"\"\"F'!\"\"" }{TEXT -1 9 ", " }{XPPEDIT 18 0 "y = D[y]/D;" "6#/%\"yG*&&%\"DG6#F$\"\"\"F'!\"\"" }{TEXT -1 2 ", " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "D = ``;" "6#/%\"DG%!G" }{TEXT -1 4 "det " }{XPPEDIT 18 0 "matrix([[a[1], b[1]], [a[2], b[2]]])" "6#-%'matrixG6#7$7$&%\"aG6#\" \"\"&%\"bG6#F+7$&F)6#\"\"#&F-6#F2" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "D[ x] = ``;" "6#/&%\"DG6#%\"xG%!G" }{TEXT -1 4 "det " }{XPPEDIT 18 0 "mat rix([[c[1], b[1]], [c[2], b[2]]])" "6#-%'matrixG6#7$7$&%\"cG6#\"\"\"&% \"bG6#F+7$&F)6#\"\"#&F-6#F2" }{TEXT -1 8 ", and " }{XPPEDIT 18 0 "D[ y] = ``;" "6#/&%\"DG6#%\"yG%!G" }{TEXT -1 4 "det " }{XPPEDIT 18 0 "mat rix([[a[1], c[1]], [a[2], c[2]]])" "6#-%'matrixG6#7$7$&%\"aG6#\"\"\"&% \"cG6#F+7$&F)6#\"\"#&F-6#F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Solutions exist provided that D " }{XPPEDIT 18 0 "``<>0" "6#0%!G\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "D[x];" "6#&%\"DG6#%\"xG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "D[y];" "6#&%\"DG6#%\"yG" }{TEXT -1 77 " are obtained by replacing the first and second columns respective ly of D by " }{XPPEDIT 18 0 "matrix([[c[1]], [c[2]]]);" "6#-%'matrixG6 #7$7#&%\"cG6#\"\"\"7#&F)6#\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 \+ " }}{PARA 0 "" 0 "" {TEXT 265 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 38 "Solve the system of linear equations: " } {XPPEDIT 18 0 "PIECEWISE([3*x-2*y=9, ``],[3*y-x=3, ``])" "6#-%*PIECEWI SEG6$7$/,&*&\"\"$\"\"\"%\"xGF+F+*&\"\"#F+%\"yGF+!\"\"\"\"*%!G7$/,&*&F* F+F/F+F+F,F0F*F2" }{TEXT -1 18 "by Cramer's rule. " }}{PARA 0 "" 0 "" {TEXT 266 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 39 "First write each equation in the form " }{XPPEDIT 18 0 "a*x+b*y=c" " 6#/,&*&%\"aG\"\"\"%\"xGF'F'*&%\"bGF'%\"yGF'F'%\"cG" }{TEXT -1 11 " to \+ obtain " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE( [3*x-2*y = 9, ``],[-x+3*y = 3, ``])" "6#-%*PIECEWISEG6$7$/,&*&\"\"$\" \"\"%\"xGF+F+*&\"\"#F+%\"yGF+!\"\"\"\"*%!G7$/,&F,F0*&F*F+F/F+F+F*F2" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "The matrix form of this system is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matri x([[3,-2],[-1,3]])*matrix([[x],[y]])=matrix([[9],[3]])" "6#/*&-%'matri xG6#7$7$\"\"$,$\"\"#!\"\"7$,$\"\"\"F-F*F0-F&6#7$7#%\"xG7#%\"yGF0-F&6#7 $7#\"\"*7#F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 45 "The determinant of the coefficient matrix is " }{XPPEDIT 18 0 "` D`=3*`.`*3-(-1)*`.`*(-2)" "6#/%#~DG,&*(\"\"$\" \"\"%\".GF(F'F(F(*(,$F(!\"\"F(F)F(,$\"\"#F,F(F," }{XPPEDIT 18 0 "``=7 " "6#/%!G\"\"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "The so lutions are: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x=d et*matrix([[9, -2], [3, 3]])/D" "6#/%\"xG*(%$detG\"\"\"-%'matrixG6#7$7 $\"\"*,$\"\"#!\"\"7$\"\"$F2F'%\"DGF0" }{TEXT -1 12 ", and " } {XPPEDIT 18 0 "y = det*matrix([[3, 9], [-1, 3]])/D" "6#/%\"yG*(%$detG \"\"\"-%'matrixG6#7$7$\"\"$\"\"*7$,$F'!\"\"F-F'%\"DGF1" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=33/7" "6#/%\"xG*&\"#L\"\"\"\"\"(!\"\"" } {TEXT -1 8 ", and " }{XPPEDIT 18 0 "y=18/7" "6#/%\"yG*&\"#=\"\"\"\" \"(!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" } {TEXT -1 29 ": The matrix multiplication: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[3, -2], [-1, 3]])*matrix([[33/7], [1 8/7]]) = matrix([[9], [3]])" "6#/*&-%'matrixG6#7$7$\"\"$,$\"\"#!\"\"7$ ,$\"\"\"F-F*F0-F&6#7$7#*&\"#LF0\"\"(F-7#*&\"#=F0F7F-F0-F&6#7$7#\"\"*7# F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "checks the solution s. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 \+ " }}{PARA 0 "" 0 "" {TEXT 263 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 39 "Solve the system of linear equations: " } {XPPEDIT 18 0 "PIECEWISE([x/sqrt(3)+y = 1, ``],[x-sqrt(3)*y = 0, ``]); " "6#-%*PIECEWISEG6$7$/,&*&%\"xG\"\"\"-%%sqrtG6#\"\"$!\"\"F+%\"yGF+F+% !G7$/,&F*F+*&-F-6#F/F+F1F+F0\"\"!F2" }{TEXT -1 18 "by Cramer's rule. \+ " }}{PARA 0 "" 0 "" {TEXT 264 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 34 "The matrix form of the system is: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1/sqrt(3), 1], [1, -sqr t(3)]])*matrix([[x], [y]]) = matrix([[1], [0]]);" "6#/*&-%'matrixG6#7$ 7$*&\"\"\"F+-%%sqrtG6#\"\"$!\"\"F+7$F+,$-F-6#F/F0F+-F&6#7$7#%\"xG7#%\" yGF+-F&6#7$7#F+7#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The determinant of the coefficient m atrix is " }{XPPEDIT 18 0 "` D` = -2;" "6#/%#~DG,$\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "The solutions are: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x = det*matrix([[1, 1], [0 , -sqrt(3)]])/D;" "6#/%\"xG*(%$detG\"\"\"-%'matrixG6#7$7$F'F'7$\"\"!,$ -%%sqrtG6#\"\"$!\"\"F'%\"DGF4" }{XPPEDIT 18 0 "`` = (-sqrt(3))/(-2);" "6#/%!G*&,$-%%sqrtG6#\"\"$!\"\"\"\"\",$\"\"#F+F+" }{TEXT -1 13 ", a nd " }{XPPEDIT 18 0 "y = det*matrix([[1/sqrt(3), 1], [1, 0]])/D;" "6#/%\"yG*(%$detG\"\"\"-%'matrixG6#7$7$*&F'F'-%%sqrtG6#\"\"$!\"\"F'7$F '\"\"!F'%\"DGF2" }{XPPEDIT 18 0 "`` = (-1)/(-2);" "6#/%!G*&,$\"\"\"!\" \"F',$\"\"#F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that i s, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=sqrt(3)/2" " 6#/%\"xG*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 8 ", and " } {XPPEDIT 18 0 "y=1/2" "6#/%\"yG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 29 ": The matrix multip lication: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix( [[1/sqrt(3), 1], [1, -sqrt(3)]])*matrix([[sqrt(3)/2], [1/2]]) = matrix ([[1], [0]]);" "6#/*&-%'matrixG6#7$7$*&\"\"\"F+-%%sqrtG6#\"\"$!\"\"F+7 $F+,$-F-6#F/F0F+-F&6#7$7#*&-F-6#F/F+\"\"#F07#*&F+F+F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 80 "Cramer's rule for the solution o f a system of 3 linear equations in 3 variables " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 25 "The system o f equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIEC EWISE([a[1]*x+b[1]*y+c[1]*z = d[1], ``],[a[2]*x+b[2]*y+c[2]*z =d[2], ` `],[a[3]*x+b[3]*y+c[3]*z =d[3], ``])" "6#-%*PIECEWISEG6%7$/,(*&&%\"aG6 #\"\"\"F-%\"xGF-F-*&&%\"bG6#F-F-%\"yGF-F-*&&%\"cG6#F-F-%\"zGF-F-&%\"dG 6#F-%!G7$/,(*&&F+6#\"\"#F-F.F-F-*&&F16#FCF-F3F-F-*&&F66#FCF-F8F-F-&F:6 #FCF<7$/,(*&&F+6#\"\"$F-F.F-F-*&&F16#FRF-F3F-F-*&&F66#FRF-F8F-F-&F:6#F RF<" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 27 "which has the matrix form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[a[1], b[1], c[1]], [a[2], b[2], c[2] ], [a[3], b[3], c[3]]])*matrix([[x], [y], [z]]) = matrix([[d[1]], [d[2 ]], [d[3]]]);" "6#/*&-%'matrixG6#7%7%&%\"aG6#\"\"\"&%\"bG6#F-&%\"cG6#F -7%&F+6#\"\"#&F/6#F7&F26#F77%&F+6#\"\"$&F/6#F?&F26#F?F--F&6#7%7#%\"xG7 #%\"yG7#%\"zGF--F&6#7%7#&%\"dG6#F-7#&FR6#F77#&FR6#F?" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 19 "has the solutions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = D[x]/D;" "6#/%\"xG*&&%\"DG6#F$ \"\"\"F'!\"\"" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "y = D[y]/D;" "6#/%\" yG*&&%\"DG6#F$\"\"\"F'!\"\"" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "z = D[ z]/D;" "6#/%\"zG*&&%\"DG6#F$\"\"\"F'!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "D = ``;" "6#/%\"DG%!G" }{TEXT -1 6 " det " }{XPPEDIT 18 0 "matr ix([[a[1], b[1], c[1]], [a[2], b[2], c[2]], [a[3], b[3], c[3]]])" "6#- %'matrixG6#7%7%&%\"aG6#\"\"\"&%\"bG6#F+&%\"cG6#F+7%&F)6#\"\"#&F-6#F5&F 06#F57%&F)6#\"\"$&F-6#F=&F06#F=" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "D[x ] = ``;" "6#/&%\"DG6#%\"xG%!G" }{TEXT -1 5 "det " }{XPPEDIT 18 0 "mat rix([[d[1], b[1], c[1]], [d[2], b[2], c[2]], [d[3], b[3], c[3]]])" "6# -%'matrixG6#7%7%&%\"dG6#\"\"\"&%\"bG6#F+&%\"cG6#F+7%&F)6#\"\"#&F-6#F5& F06#F57%&F)6#\"\"$&F-6#F=&F06#F=" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "D[ y] = ``;" "6#/&%\"DG6#%\"yG%!G" }{TEXT -1 5 "det " }{XPPEDIT 18 0 "ma trix([[a[1], d[1], c[1]], [a[2], d[2], c[2]], [a[3], d[3], c[3]]])" "6 #-%'matrixG6#7%7%&%\"aG6#\"\"\"&%\"dG6#F+&%\"cG6#F+7%&F)6#\"\"#&F-6#F5 &F06#F57%&F)6#\"\"$&F-6#F=&F06#F=" }{TEXT -1 8 ", and " }{XPPEDIT 18 0 "D[z] = ``;" "6#/&%\"DG6#%\"zG%!G" }{TEXT -1 5 "det " }{XPPEDIT 18 0 "matrix([[a[1], b[1], d[1]], [a[2], b[2], d[2]], [a[3], b[3], d[3 ]]])" "6#-%'matrixG6#7%7%&%\"aG6#\"\"\"&%\"bG6#F+&%\"dG6#F+7%&F)6#\"\" #&F-6#F5&F06#F57%&F)6#\"\"$&F-6#F=&F06#F=" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "Solutions ex ist provided that D" }{XPPEDIT 18 0 "``<>0" "6#0%!G\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "D[x], D [y]" "6$&%\"DG6#%\"xG&F$6#%\"yG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "D [z]" "6#&%\"DG6#%\"zG" }{TEXT -1 83 " are obtained by replacing the fi rst second and third columns respectively of D by " }{XPPEDIT 18 0 "ma trix([[d[1]],[d[2]],[d[3]]])" "6#-%'matrixG6#7%7#&%\"dG6#\"\"\"7#&F)6# \"\"#7#&F)6#\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{PARA 0 "" 0 "" {TEXT 267 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 38 "Solve the system of linear equat ions: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE( [2*x+y-z=2,``],[4*x-y-3*z=-2,``],[2*x+2*y+z=9,``])" "6#-%*PIECEWISEG6% 7$/,(*&\"\"#\"\"\"%\"xGF+F+%\"yGF+%\"zG!\"\"F*%!G7$/,(*&\"\"%F+F,F+F+F -F/*&\"\"$F+F.F+F/,$F*F/F07$/,(*&F*F+F,F+F+*&F*F+F-F+F+F.F+\"\"*F0" } {TEXT -1 13 "------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 18 "by Cramer's rule. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 36 "The matri x form of this system is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[2,1,-1],[4,-1,-3],[2,2,1]])*matrix([[x],[y],[z ]])=matrix([[2],[-2],[9]])" "6#/*&-%'matrixG6#7%7%\"\"#\"\"\",$F+!\"\" 7%\"\"%,$F+F-,$\"\"$F-7%F*F*F+F+-F&6#7%7#%\"xG7#%\"yG7#%\"zGF+-F&6#7%7 #F*7#,$F*F-7#\"\"*" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{GLPLOT2D 278 177 177 {PLOTDATA 2 "6B-%'CURVESG6$7$7$$!3A+++++++S!# =$!35+++++++?F*7$F($\"33+++++++@!#<-%'COLOURG6&%$RGBG\"\"!F5F5-F$6$7$7 $$\"3!**************R#F0F+7$F:F.F1-F$6(7$7$$!\"\"!\"#FA7$$\"$q#FCFE7%7 $$\"+\"*pY6D!\"*$\"+4I`FCFA7$$\"$q%FCFE7%7$$\"+\"*pY6XFKFLF`p7$$\"+4I`9&FKF^q7$$\"+'*Rk6DF K$!*C%41iFKFO-FT6&F4FX$\"\"$FBF_rFZFhn-F$6(7$7$F^oF^p7$FaoF_q7%7$$\"+/ gNFCFA7$$\"$q%FCFE7%7$$\"+\"*pY6XFKFLF`p7$$\"+4I`9&FKF^q7$$\"+'*Rk6DFK$ !*C%41iFKFO-FT6&F4FX$\"\"$FBF_rFZFhn-F$6(7$7$F^oF^p7$FaoF_q7%7$$\"+/gN FCFA7$$\"$q%FCFE7%7$$\"+ \"*pY6XFKFLF`p7$$\"+4I`9&FKF^q7$$\"+'*Rk6DFK$!*C%41iFKFO-FT6&F4FX$\"\"$FBF_rF ZFhn-F$6(7$7$F^oF^p7$FaoF_q7%7$$\"+/gNFCFA7$$\"$q%FCFE7%7$$\"+\"*pY6XFKFLF`p7$$\"+4 I`9&FKF ^q7$$\"+'*Rk6DFK$!*C%41iFKFO-FT6&F4FX$\"\"$FBF_rFZFhn-F$6(7$7$F^oF^p7$ FaoF_q7%7$$\"+/gN " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 65 "Solve the following systems of equations by using Cramer' s rule. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "PIECEWISE([2*x-y = -11, ``],[x+3*y = 12, \+ ``])" "6#-%*PIECEWISEG6$7$/,&*&\"\"#\"\"\"%\"xGF+F+%\"yG!\"\",$\"#6F.% !G7$/,&F,F+*&\"\"$F+F-F+F+\"#7F1" }{TEXT -1 32 " \+ (b) " }{XPPEDIT 18 0 "PIECEWISE([3*x-2*y = -4, ``],[-5*x+4*y = - 1, ``]);" "6#-%*PIECEWISEG6$7$/,&*&\"\"$\"\"\"%\"xGF+F+*&\"\"#F+%\"yGF +!\"\",$\"\"%F0%!G7$/,&*&\"\"&F+F,F+F0*&F2F+F/F+F+,$F+F0F3" }{TEXT -1 2 " " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 "(c) " }{XPPEDIT 18 0 "PIECEWISE([x/2-y/3 = 4, ``],[x/4+y/2 = 6, `` ]);" "6#-%*PIECEWISEG6$7$/,&*&%\"xG\"\"\"\"\"#!\"\"F+*&%\"yGF+\"\"$F-F -\"\"%%!G7$/,&*&F*F+F1F-F+*&F/F+F,F-F+\"\"'F2" }{TEXT -1 32 " \+ (d) " }{XPPEDIT 18 0 "PIECEWISE([sqrt(2)*x+sqrt(3)* y = 4, ``],[sqrt(18)*x-sqrt(12)*y = -3, ``]);" "6#-%*PIECEWISEG6$7$/,& *&-%%sqrtG6#\"\"#\"\"\"%\"xGF.F.*&-F+6#\"\"$F.%\"yGF.F.\"\"%%!G7$/,&*& -F+6#\"#=F.F/F.F.*&-F+6#\"#7F.F4F.!\"\",$F3FBF6" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "An s " }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "x = -3, y = 5 " "6$/%\"xG,$\"\"$!\"\"/%\"yG\"\"&" }{TEXT -1 7 " (b) " }{XPPEDIT 18 0 "x = -9, y = -23/2" "6$/%\"xG,$\"\"*!\"\"/%\"yG,$*&\"#B\"\"\"\"\" #F'F'" }{TEXT -1 7 " (c) " }{XPPEDIT 18 0 "x = 12, y = 6" "6$/%\"xG \"#7/%\"yG\"\"'" }{TEXT -1 7 " (d) " }{XPPEDIT 18 0 "x = sqrt(2)/2, \+ y = sqrt(3)" "6$/%\"xG*&-%%sqrtG6#\"\"#\"\"\"F)!\"\"/%\"yG-F'6#\"\"$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "\{2*x-y = -11,x+3*y = 12\};\nsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/,&*&\"\"#\"\"\"%\"xGF(F(%\"yG!\"\"!#6/,&F )F(*&\"\"$F(F*F(F(\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG!\"$ /%\"yG\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "\{3*x-2*y = -4,-5*x+4*y = -1\};\nsolve(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<$/,&*&\"\"&\"\"\"%\"xGF(!\"\"*&\"\"%F (%\"yGF(F(F*/,&*&\"\"$F(F)F(F(*&\"\"#F(F-F(F*!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG!\"*/%\"yG#!#B\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "\{x/2-y/3 = 4,x/4+ y/2 = 6\};\nsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/,&*&\"\"#! \"\"%\"xG\"\"\"F**&\"\"$F(%\"yGF*F(\"\"%/,&*&F.F(F)F*F**&F'F(F-F*F*\" \"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG\"#7/%\"yG\"\"'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "\{sqrt(2)*x+sqrt(3)*y = 4,sq rt(18)*x-sqrt(12)*y = -3\};\nsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/,&*&\"\"##\"\"\"F'%\"xGF)F)*&\"\"$F(%\"yGF)F)\"\"%/,&*(F,F)F' F(F*F)F)*(F'F)F,F(F-F)!\"\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/ %\"yG*$\"\"$#\"\"\"\"\"#/%\"xG,$*&F*!\"\"F*F(F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 65 "Solve the following systems of equat ions by using Cramer's rule. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "PIECEWISE([x+y+z \+ = 6, ``],[x-y+z = 2, ``],[2*x+y+z = 7, ``]);" "6#-%*PIECEWISEG6%7$/,(% \"xG\"\"\"%\"yGF*%\"zGF*\"\"'%!G7$/,(F)F*F+!\"\"F,F*\"\"#F.7$/,(*&F3F* F)F*F*F+F*F,F*\"\"(F." }{TEXT -1 31 " (b) " }{XPPEDIT 18 0 "PIECEWISE([x+2*y=8, ``],[x-3*y+z = -2, ``],[2*x-y = 1, ``])" "6#-%*PIECEWISEG6%7$/,&%\"xG\"\"\"*&\"\"#F*%\"yGF*F*\"\")%!G7$/ ,(F)F**&\"\"$F*F-F*!\"\"%\"zGF*,$F,F5F/7$/,&*&F,F*F)F*F*F-F5F*F/" } {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 "(c) " }{XPPEDIT 18 0 "PIECEWISE([2*x-2*y+3*z = 7, ``],[x+ y-z = -2, ``],[3*x+y-2*z = 5, ``]);" "6#-%*PIECEWISEG6%7$/,(*&\"\"#\" \"\"%\"xGF+F+*&F*F+%\"yGF+!\"\"*&\"\"$F+%\"zGF+F+\"\"(%!G7$/,(F,F+F.F+ F2F/,$F*F/F47$/,(*&F1F+F,F+F+F.F+*&F*F+F2F+F/\"\"&F4" }{TEXT -1 25 " \+ (d) " }{XPPEDIT 18 0 "PIECEWISE([2*x-3*y+z = 1, ``] ,[x+4*y-z = 0, ``],[3*x-y+2*z = 0, ``])" "6#-%*PIECEWISEG6%7$/,(*&\"\" #\"\"\"%\"xGF+F+*&\"\"$F+%\"yGF+!\"\"%\"zGF+F+%!G7$/,(F,F+*&\"\"%F+F/F +F+F1F0\"\"!F27$/,(*&F.F+F,F+F+F/F0*&F*F+F1F+F+F8F2" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "A ns " }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "x=1, y=2, z \+ =3" "6%/%\"xG\"\"\"/%\"yG\"\"#/%\"zG\"\"$" }{TEXT -1 7 " (b) " } {XPPEDIT 18 0 "x = 2, y = 3, z = 5" "6%/%\"xG\"\"#/%\"yG\"\"$/%\"zG\" \"&" }{TEXT -1 6 " (c) " }{XPPEDIT 18 0 "x = 5/3, y = -22/3, z = -11/ 3" "6%/%\"xG*&\"\"&\"\"\"\"\"$!\"\"/%\"yG,$*&\"#AF'F(F)F)/%\"zG,$*&\"# 6F'F(F)F)" }{TEXT -1 6 " (d) " }{XPPEDIT 18 0 "x = 7/16, y = -5/16, z = -13/16" "6%/%\"xG*&\"\"(\"\"\"\"#;!\"\"/%\"yG,$*&\"\"&F'F(F)F)/%\"z G,$*&\"#8F'F(F)F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "\{x+y+z=6,x-y+z = 2,2*x+y+z \+ = 7\};\nsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/,(%\"xG\"\"\"% \"yG!\"\"%\"zGF'\"\"#/,(F&F'F(F'F*F'\"\"'/,(*&F+F'F&F'F'F(F'F*F'\"\"( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"zG\"\"$/%\"yG\"\"#/%\"xG\"\" \"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "\{x+2*y=8,x-3*y+z = -2,2*x-y = 1\};\nsolve(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<%/,&%\"xG\"\"\"*&\"\"#F'%\"yGF'F'\"\" )/,(F&F'*&\"\"$F'F*F'!\"\"%\"zGF'!\"#/,&*&F)F'F&F'F'F*F0F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"yG\"\"$/%\"xG\"\"#/%\"zG\"\"&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "\{2*x-2*y+3*z = 7,x+y-z = -2,3*x+y-2*z = 5\};\nsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/,(*&\"\"#\"\"\"%\"xGF(F(*&F'F(%\"yGF(!\" \"*&\"\"$F(%\"zGF(F(\"\"(/,(F)F(F+F(F/F,!\"#/,(*&F.F(F)F(F(F+F(*&F'F(F /F(F,\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"yG#!#A\"\"$/%\"zG# !#6F(/%\"xG#\"\"&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "\{2* x-3*y+z = 1,x+4*y-z = 0,3*x-y+2*z = 0\};\nsolve(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<%/,(*&\"\"#\"\"\"%\"xGF(F(*&\"\"$F(%\"yGF(!\"\"%\"zG F(F(/,(F)F(*&\"\"%F(F,F(F(F.F-\"\"!/,(*&F+F(F)F(F(F,F-*&F'F(F.F(F(F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"xG#\"\"(\"#;/%\"yG#!\"&F(/%\"z G#!#8F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "__________________ _________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "_______________________________ ____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 18 "Code for pictures " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 257 "" 0 " " {TEXT -1 50 "Code for diagonal expansion of 3 by 3 determinant " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1545 "A := matrix([[a,b,c],[d,e,f],[g,h,k]]);\np1 := plot([[[-.4,-.2], [-.4,2.1]],[[2.4,-.2],[2.4,2.1]]],color=black):\nt1 := plots[textplot] ([[0,2,A[1,1]],[1,2,A[1,2]],[2,2,A[1,3]],\n [0,1,A[2,1]],[1,1,A[2,2]] ,[2,1,A[2,3]],\n [0,0,A[3,1]],[1,0,A[3,2]],[2,0,A[3,3]]],font=[TIMES, ITALIC,12],color=black):\nt2 := plots[textplot]([[3,2,A[1,1]],[4,2,A[1 ,2]],[3,1,A[2,1]],[4,1,A[2,2]],\n [3,0,A[3,1]],[4,0,A[3,2]]],font=[TIM ES,ITALIC,12],color=COLOR(RGB,0,.6,0)):\nt3 := plots[textplot]([[3,3,A [3,1]*A[2,2]*A[1,3]],[4,3,A[3,2]*A[2,3]*A[1,1]],\n [5,3,A[3,3]* A[2,1 ]*A[1,2]]],\n font=[TIMES,ITALIC,12],color=COLOR(RGB,.3,0,1)):\nt4 := plots[textplot]([[3,-1,A[1,1]*A[2,2]*A[3,3]],[4,-1,A[1,2]*A[2,3]*A[3, 1]],\n [5,-1,A[1,3]*A[2,1]*A[3,2]]],\n font=[TIMES,ITALIC,12],color= COLOR(RGB,1,0,.3)):\np2 := plottools[arrow]([-.01,-.01],[2.7,2.7],0,.1 5,.05,arrow,\n color=COLOR(RGB,.6,.6,1),thickness=2,linestyle=2): \np3 := plottools[arrow]([.99,-.01],[3.7,2.7],0,.15,.05,arrow,\n c olor=COLOR(RGB,.6,.6,1),thickness=2,linestyle=2):\np4 := plottools[arr ow]([1.99,-.01],[4.7,2.7],0,.15,.05,arrow,\n color=COLOR(RGB,.6,.6 ,1),thickness=2,linestyle=2):\np5 := plottools[arrow]([-.01,1.99],[2.7 ,-.702],0,.15,.05,arrow,\n color=COLOR(RGB,1,.3,.3),thickness=2,li nestyle=2):\np6 := plottools[arrow]([.99,1.99],[3.7,-.702],0,.15,.05,a rrow,\n color=COLOR(RGB,1,.3,.3),thickness=2,linestyle=2):\np7 := \+ plottools[arrow]([1.99,1.99],[4.7,-.702],0,.15,.05,arrow,\n color= COLOR(RGB,1,.3,.3),thickness=2,linestyle=2):\nplots[display]([p1,p2,p3 ,p4,p5,p6,p7,t1,t2,t3,t4],axes=none);" }}}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[2,1,- 1],[4,-1,-3],[2,2,1]])*matrix([[x],[y],[z]])=matrix([[2],[-2],[9]])" " 6#/*&-%'matrixG6#7%7%\"\"#\"\"\",$F+!\"\"7%\"\"%,$F+F-,$\"\"$F-7%F*F*F +F+-F&6#7%7#%\"xG7#%\"yG7#%\"zGF+-F&6#7%7#F*7#,$F*F-7#\"\"*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1544 "A := matrix([[2,1,-1],[4,-1,-3],[2,2,1]]);\np1 := p lot([[[-.4,-.2],[-.4,2.1]],[[2.4,-.2],[2.4,2.1]]],color=black):\nt1 := plots[textplot]([[0,2,A[1,1]],[1,2,A[1,2]],[2,2,A[1,3]],\n [0,1,A[2, 1]],[1,1,A[2,2]],[2,1,A[2,3]],\n [0,0,A[3,1]],[1,0,A[3,2]],[2,0,A[3,3 ]]],font=[TIMES,ROMAN,12],color=black):\nt2 := plots[textplot]([[3,2,A [1,1]],[4,2,A[1,2]],[3,1,A[2,1]],[4,1,A[2,2]],\n [3,0,A[3,1]],[4,0,A[3 ,2]]],font=[TIMES,ROMAN,12],color=COLOR(RGB,0,.6,0)):\nt3 := plots[tex tplot]([[3,3,A[3,1]*A[2,2]*A[1,3]],[4,3,A[3,2]*A[2,3]*A[1,1]],\n [5,3 ,A[3,3]* A[2,1]*A[1,2]]],\n font=[TIMES,ROMAN,12],color=COLOR(RGB,.3, 0,1)):\nt4 := plots[textplot]([[3,-1,A[1,1]*A[2,2]*A[3,3]],[4,-1,A[1,2 ]*A[2,3]*A[3,1]],\n [5,-1,A[1,3]*A[2,1]*A[3,2]]],\n font=[TIMES,ROMA N,12],color=COLOR(RGB,1,0,.3)):\np2 := plottools[arrow]([-.01,-.01],[2 .7,2.7],0,.15,.05,arrow,\n color=COLOR(RGB,.6,.6,1),thickness=2,li nestyle=2):\np3 := plottools[arrow]([.99,-.01],[3.7,2.7],0,.15,.05,arr ow,\n color=COLOR(RGB,.6,.6,1),thickness=2,linestyle=2):\np4 := pl ottools[arrow]([1.99,-.01],[4.7,2.7],0,.15,.05,arrow,\n color=COLO R(RGB,.6,.6,1),thickness=2,linestyle=2):\np5 := plottools[arrow]([-.01 ,1.99],[2.7,-.702],0,.15,.05,arrow,\n color=COLOR(RGB,1,.3,.3),thi ckness=2,linestyle=2):\np6 := plottools[arrow]([.99,1.99],[3.7,-.702], 0,.15,.05,arrow,\n color=COLOR(RGB,1,.3,.3),thickness=2,linestyle= 2):\np7 := plottools[arrow]([1.99,1.99],[4.7,-.702],0,.15,.05,arrow,\n color=COLOR(RGB,1,.3,.3),thickness=2,linestyle=2):\nplots[display ]([p1,p2,p3,p4,p5,p6,p7,t1,t2,t3,t4],axes=none);" }}}{PARA 257 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "mat rix([[2,1,-1],[4,-1,-3],[2,2,1]])*matrix([[x],[y],[z]])=matrix([[2],[- 2],[9]])" "6#/*&-%'matrixG6#7%7%\"\"#\"\"\",$F+!\"\"7%\"\"%,$F+F-,$\" \"$F-7%F*F*F+F+-F&6#7%7#%\"xG7#%\"yG7#%\"zGF+-F&6#7%7#F*7#,$F*F-7#\"\" *" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1545 "A := matrix([[2,1,-1],[-2,-1,-3],[9,2,1]]) ;\np1 := plot([[[-.4,-.2],[-.4,2.1]],[[2.4,-.2],[2.4,2.1]]],color=blac k):\nt1 := plots[textplot]([[0,2,A[1,1]],[1,2,A[1,2]],[2,2,A[1,3]],\n \+ [0,1,A[2,1]],[1,1,A[2,2]],[2,1,A[2,3]],\n [0,0,A[3,1]],[1,0,A[3,2]], [2,0,A[3,3]]],font=[TIMES,ROMAN,12],color=black):\nt2 := plots[textplo t]([[3,2,A[1,1]],[4,2,A[1,2]],[3,1,A[2,1]],[4,1,A[2,2]],\n [3,0,A[3,1] ],[4,0,A[3,2]]],font=[TIMES,ROMAN,12],color=COLOR(RGB,0,.6,0)):\nt3 := plots[textplot]([[3,3,A[3,1]*A[2,2]*A[1,3]],[4,3,A[3,2]*A[2,3]*A[1,1] ],\n [5,3,A[3,3]* A[2,1]*A[1,2]]],\n font=[TIMES,ROMAN,12],color=COL OR(RGB,.3,0,1)):\nt4 := plots[textplot]([[3,-1,A[1,1]*A[2,2]*A[3,3]],[ 4,-1,A[1,2]*A[2,3]*A[3,1]],\n [5,-1,A[1,3]*A[2,1]*A[3,2]]],\n font=[ TIMES,ROMAN,12],color=COLOR(RGB,1,0,.3)):\np2 := plottools[arrow]([-.0 1,-.01],[2.7,2.7],0,.15,.05,arrow,\n color=COLOR(RGB,.6,.6,1),thic kness=2,linestyle=2):\np3 := plottools[arrow]([.99,-.01],[3.7,2.7],0,. 15,.05,arrow,\n color=COLOR(RGB,.6,.6,1),thickness=2,linestyle=2): \np4 := plottools[arrow]([1.99,-.01],[4.7,2.7],0,.15,.05,arrow,\n \+ color=COLOR(RGB,.6,.6,1),thickness=2,linestyle=2):\np5 := plottools[ar row]([-.01,1.99],[2.7,-.702],0,.15,.05,arrow,\n color=COLOR(RGB,1, .3,.3),thickness=2,linestyle=2):\np6 := plottools[arrow]([.99,1.99],[3 .7,-.702],0,.15,.05,arrow,\n color=COLOR(RGB,1,.3,.3),thickness=2, linestyle=2):\np7 := plottools[arrow]([1.99,1.99],[4.7,-.702],0,.15,.0 5,arrow,\n color=COLOR(RGB,1,.3,.3),thickness=2,linestyle=2):\nplo ts[display]([p1,p2,p3,p4,p5,p6,p7,t1,t2,t3,t4],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[2,1,-1],[4,-1,-3],[2,2,1]])*matrix([[x],[y],[z]])=matri x([[2],[-2],[9]])" "6#/*&-%'matrixG6#7%7%\"\"#\"\"\",$F+!\"\"7%\"\"%,$ F+F-,$\"\"$F-7%F*F*F+F+-F&6#7%7#%\"xG7#%\"yG7#%\"zGF+-F&6#7%7#F*7#,$F* F-7#\"\"*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1544 "A := matrix([[2,2,-1],[4,-2,-3],[ 2,9,1]]);\np1 := plot([[[-.4,-.2],[-.4,2.1]],[[2.4,-.2],[2.4,2.1]]],co lor=black):\nt1 := plots[textplot]([[0,2,A[1,1]],[1,2,A[1,2]],[2,2,A[1 ,3]],\n [0,1,A[2,1]],[1,1,A[2,2]],[2,1,A[2,3]],\n [0,0,A[3,1]],[1,0, A[3,2]],[2,0,A[3,3]]],font=[TIMES,ROMAN,12],color=black):\nt2 := plots [textplot]([[3,2,A[1,1]],[4,2,A[1,2]],[3,1,A[2,1]],[4,1,A[2,2]],\n [3, 0,A[3,1]],[4,0,A[3,2]]],font=[TIMES,ROMAN,12],color=COLOR(RGB,0,.6,0)) :\nt3 := plots[textplot]([[3,3,A[3,1]*A[2,2]*A[1,3]],[4,3,A[3,2]*A[2,3 ]*A[1,1]],\n [5,3,A[3,3]* A[2,1]*A[1,2]]],\n font=[TIMES,ROMAN,12],c olor=COLOR(RGB,.3,0,1)):\nt4 := plots[textplot]([[3,-1,A[1,1]*A[2,2]*A [3,3]],[4,-1,A[1,2]*A[2,3]*A[3,1]],\n [5,-1,A[1,3]*A[2,1]*A[3,2]]],\n font=[TIMES,ROMAN,12],color=COLOR(RGB,1,0,.3)):\np2 := plottools[arr ow]([-.01,-.01],[2.7,2.7],0,.15,.05,arrow,\n color=COLOR(RGB,.6,.6 ,1),thickness=2,linestyle=2):\np3 := plottools[arrow]([.99,-.01],[3.7, 2.7],0,.15,.05,arrow,\n color=COLOR(RGB,.6,.6,1),thickness=2,lines tyle=2):\np4 := plottools[arrow]([1.99,-.01],[4.7,2.7],0,.15,.05,arrow ,\n color=COLOR(RGB,.6,.6,1),thickness=2,linestyle=2):\np5 := plot tools[arrow]([-.01,1.99],[2.7,-.702],0,.15,.05,arrow,\n color=COLO R(RGB,1,.3,.3),thickness=2,linestyle=2):\np6 := plottools[arrow]([.99, 1.99],[3.7,-.702],0,.15,.05,arrow,\n color=COLOR(RGB,1,.3,.3),thic kness=2,linestyle=2):\np7 := plottools[arrow]([1.99,1.99],[4.7,-.702], 0,.15,.05,arrow,\n color=COLOR(RGB,1,.3,.3),thickness=2,linestyle= 2):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,t1,t2,t3,t4],axes=none);" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[2,1,-1],[4,-1,-3],[2,2,1]])*matrix([[x],[y],[z ]])=matrix([[2],[-2],[9]])" "6#/*&-%'matrixG6#7%7%\"\"#\"\"\",$F+!\"\" 7%\"\"%,$F+F-,$\"\"$F-7%F*F*F+F+-F&6#7%7#%\"xG7#%\"yG7#%\"zGF+-F&6#7%7 #F*7#,$F*F-7#\"\"*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1543 "A := matrix([[2,1,2],[4,- 1,-2],[2,2,9]]);\np1 := plot([[[-.4,-.2],[-.4,2.1]],[[2.4,-.2],[2.4,2. 1]]],color=black):\nt1 := plots[textplot]([[0,2,A[1,1]],[1,2,A[1,2]],[ 2,2,A[1,3]],\n [0,1,A[2,1]],[1,1,A[2,2]],[2,1,A[2,3]],\n [0,0,A[3,1] ],[1,0,A[3,2]],[2,0,A[3,3]]],font=[TIMES,ROMAN,12],color=black):\nt2 : = plots[textplot]([[3,2,A[1,1]],[4,2,A[1,2]],[3,1,A[2,1]],[4,1,A[2,2]] ,\n [3,0,A[3,1]],[4,0,A[3,2]]],font=[TIMES,ROMAN,12],color=COLOR(RGB,0 ,.6,0)):\nt3 := plots[textplot]([[3,3,A[3,1]*A[2,2]*A[1,3]],[4,3,A[3,2 ]*A[2,3]*A[1,1]],\n [5,3,A[3,3]* A[2,1]*A[1,2]]],\n font=[TIMES,ROMA N,12],color=COLOR(RGB,.3,0,1)):\nt4 := plots[textplot]([[3,-1,A[1,1]*A [2,2]*A[3,3]],[4,-1,A[1,2]*A[2,3]*A[3,1]],\n [5,-1,A[1,3]*A[2,1]*A[3, 2]]],\n font=[TIMES,ROMAN,12],color=COLOR(RGB,1,0,.3)):\np2 := plotto ols[arrow]([-.01,-.01],[2.7,2.7],0,.15,.05,arrow,\n color=COLOR(RG B,.6,.6,1),thickness=2,linestyle=2):\np3 := plottools[arrow]([.99,-.01 ],[3.7,2.7],0,.15,.05,arrow,\n color=COLOR(RGB,.6,.6,1),thickness= 2,linestyle=2):\np4 := plottools[arrow]([1.99,-.01],[4.7,2.7],0,.15,.0 5,arrow,\n color=COLOR(RGB,.6,.6,1),thickness=2,linestyle=2):\np5 \+ := plottools[arrow]([-.01,1.99],[2.7,-.702],0,.15,.05,arrow,\n col or=COLOR(RGB,1,.3,.3),thickness=2,linestyle=2):\np6 := plottools[arrow ]([.99,1.99],[3.7,-.702],0,.15,.05,arrow,\n color=COLOR(RGB,1,.3,. 3),thickness=2,linestyle=2):\np7 := plottools[arrow]([1.99,1.99],[4.7, -.702],0,.15,.05,arrow,\n color=COLOR(RGB,1,.3,.3),thickness=2,lin estyle=2):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,t1,t2,t3,t4],axes=non e);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }