document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=618509>Question 618509</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, I try to solve for a,b,c but do not know how to get them. Thank you\r<BR>\n" );
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document.write( "x,y,xy, x^2 y, n are all just numbers. thank you\r<BR>\n" );
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document.write( "Y = na+bx1+cx2<BR>\n" );
document.write( "XY = ax1+bx2+cx3<BR>\n" );
document.write( "X^2 Y = ax2+bx3+cx4<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #388983 by </B><A HREF=/tutors/aboutme.mpl?userid=swincher4391><B>swincher4391(1107)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=swincher4391><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=swincher4391><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=388983\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Rearrange in this way:\r<BR>\n" );
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document.write( "Y =    na + bx1 + cx2 + 0 + 0<BR>\n" );
document.write( "XY =    0 + ax1 + bx2 + cx3 + 0<BR>\n" );
document.write( "X^2Y =  0 + 0   + ax2 + bx3 + cx4\r<BR>\n" );
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document.write( "From here we can make a matrix.<BR>\n" );
document.write( "a  b  c <BR>\n" );
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document.write( "n  x1 x2    Y<BR>\n" );
document.write( "x1 x2 x3    XY<BR>\n" );
document.write( "x2 x3 x4    X^2 Y\r<BR>\n" );
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document.write( "Row reduce this matrix.\r<BR>\n" );
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document.write( "Divide the first row entries by n.\r<BR>\n" );
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document.write( "1   x1/n x2/n Y/n<BR>\n" );
document.write( "x1  x2   x3   XY<BR>\n" );
document.write( "x2  x3   x4   X^2Y\r<BR>\n" );
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document.write( "Add -x1 * the 1st row to the 2nd row.\r<BR>\n" );
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document.write( "1     x1/n           x2/n              Y/n<BR>\n" );
document.write( "0  (x2n -x1^2)/n    (x3n-x1x2)/n      (XYn - x1Y)/n<BR>\n" );
document.write( "x2    x3             x4                X^2Y\r<BR>\n" );
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document.write( "Add -x2 * the 1st row to the 3rd row.<BR>\n" );
document.write( "1    x1/n           x2/n             Y/n<BR>\n" );
document.write( "0  (x2n -x1^2)/n    (x3n-x1x2)/n     (XYn - x1Y)/n<BR>\n" );
document.write( "0   (x3n - x1x2)/n  (x4n -x2^2)/n    (X^2Yn - x2Y)/n\r<BR>\n" );
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document.write( "Multiply row 2 by n/(x2n-x1^2)\r<BR>\n" );
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document.write( "1         x1/n        x2/n                   Y/n<BR>\n" );
document.write( "0          1         (x3n-x1x2)/(x2n-x1^2)  (XYn-x1Y)/(x2n-x1^2)<BR>\n" );
document.write( "0   (x3n - x1x2)/n   (x4n -x2^2)/n          (X^2Yn - x2Y)/n\r<BR>\n" );
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document.write( "Add (x1x2-x3n)/n * row 2 to row 3.\r<BR>\n" );
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document.write( "1   x1/n  x2/n                                            Y/n<BR>\n" );
document.write( "0    1   (x3n-x1x2)/(x2n-x1^2)                           (XYn-x1Y)/(x2n-x1^2)<BR>\n" );
document.write( "0    0  (2x1x2x3-x3^2n + x2x4n -x1^2x4 -x2^3)/(x2n -x1^2)  ?\r<BR>\n" );
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document.write( "? =  (x1x2XY + x1x3Y + X^2Yx2n - X^2x1-x2^2Y)/(x2n -x1^2)\r<BR>\n" );
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document.write( "This is starting to get messy.\r<BR>\n" );
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document.write( "Let x1=A  x2=B x3=C x4=D  X^2Y = w  Y = y XY = z for space conservation.\r<BR>\n" );
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document.write( "1 A/n          B/n                          |        y/n<BR>\n" );
document.write( "0  1   (Cn-AB)/(Bn-A^2)                     |     (zn-Ay)/(Bn-A^2)<BR>\n" );
document.write( "0  0 (2ABC-C^2n + BDn - A^2D - B^3)/(Bn-A^2)|(ABw + ACy + Bnw - A^2w -B^2y)/(Bn-A^2)\r<BR>\n" );
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document.write( "Sorry that it wouldn't fit.\r<BR>\n" );
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document.write( "Next step is to multiply row 3 by (Bn-A^2)/ (2ABC-C^2n + BDn - A^2D - B^3)\r<BR>\n" );
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document.write( "1  A/n B/n               |      y/n<BR>\n" );
document.write( "0   1   (Cn-AB)/(Bn-A^2) |    (zn-Ay)/(Bn-A^2)<BR>\n" );
document.write( "0   0   1                |    (wAB+yAC+nwB-wA^2-yB^2)/(2ABC-nC^2+nBD-A^2D-B^3)\r<BR>\n" );
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document.write( "We're almost there!\r<BR>\n" );
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document.write( "Multiply row 3 by (AB-Cn)/(Bn-A^2) and add to row 2.\r<BR>\n" );
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document.write( "1  A/n B/n               |      y/n<BR>\n" );
document.write( "0   1   0                |      b<BR>\n" );
document.write( "0   0   1                |      c\r<BR>\n" );
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document.write( "b = (wA^2B^2-wA^3B+nwAB^2-nwABC-yA^2BC+nwA^2C-nzB^3 + yA^3D-n^2wBC+nyB^2C+2nzABC-nzA^2D-nyABD+n^2zBD-n^2zC^2)/(2ABC-nC^2+nBD-A^2D-B^3)(nB-A^2)<BR>\n" );
document.write( "We've solved for b and c, but now we have to find a.\r<BR>\n" );
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document.write( "Take row 3 * (-B/n)  and add to row 1.\r<BR>\n" );
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document.write( "1  A/n  0  |(-wAB^2+wA^2B-nwB^2+yABC-yA^2D+nyBD-nyC^2)/(n(2ABC-nC^2+nBD-A^2D-B^3))<BR>\n" );
document.write( "0   1   0  | b<BR>\n" );
document.write( "0   0   1  | c\r<BR>\n" );
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document.write( "One more step!\r<BR>\n" );
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document.write( "Take row 2 * (-A/n) and add it to row 1.\r<BR>\n" );
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document.write( "Take a stab at finishing the rest.\r<BR>\n" );
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document.write( "That was a doozy!\r<BR>\n" );
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document.write( "Hope this helped!\r<BR>\n" );
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