Question 618509: Hi, I try to solve for a,b,c but do not know how to get them. Thank you
x,y,xy, x^2 y, n are all just numbers. thank you
Y = na+bx1+cx2
XY = ax1+bx2+cx3
X^2 Y = ax2+bx3+cx4
Answer by swincher4391(1107)   (Show Source):
You can put this solution on YOUR website! Rearrange in this way:
Y = na + bx1 + cx2 + 0 + 0
XY = 0 + ax1 + bx2 + cx3 + 0
X^2Y = 0 + 0 + ax2 + bx3 + cx4
From here we can make a matrix.
a b c
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n x1 x2 Y
x1 x2 x3 XY
x2 x3 x4 X^2 Y
Row reduce this matrix.
Divide the first row entries by n.
1 x1/n x2/n Y/n
x1 x2 x3 XY
x2 x3 x4 X^2Y
Add -x1 * the 1st row to the 2nd row.
1 x1/n x2/n Y/n
0 (x2n -x1^2)/n (x3n-x1x2)/n (XYn - x1Y)/n
x2 x3 x4 X^2Y
Add -x2 * the 1st row to the 3rd row.
1 x1/n x2/n Y/n
0 (x2n -x1^2)/n (x3n-x1x2)/n (XYn - x1Y)/n
0 (x3n - x1x2)/n (x4n -x2^2)/n (X^2Yn - x2Y)/n
Multiply row 2 by n/(x2n-x1^2)
1 x1/n x2/n Y/n
0 1 (x3n-x1x2)/(x2n-x1^2) (XYn-x1Y)/(x2n-x1^2)
0 (x3n - x1x2)/n (x4n -x2^2)/n (X^2Yn - x2Y)/n
Add (x1x2-x3n)/n * row 2 to row 3.
1 x1/n x2/n Y/n
0 1 (x3n-x1x2)/(x2n-x1^2) (XYn-x1Y)/(x2n-x1^2)
0 0 (2x1x2x3-x3^2n + x2x4n -x1^2x4 -x2^3)/(x2n -x1^2) ?
? = (x1x2XY + x1x3Y + X^2Yx2n - X^2x1-x2^2Y)/(x2n -x1^2)
This is starting to get messy.
Let x1=A x2=B x3=C x4=D X^2Y = w Y = y XY = z for space conservation.
1 A/n B/n | y/n
0 1 (Cn-AB)/(Bn-A^2) | (zn-Ay)/(Bn-A^2)
0 0 (2ABC-C^2n + BDn - A^2D - B^3)/(Bn-A^2)|(ABw + ACy + Bnw - A^2w -B^2y)/(Bn-A^2)
Sorry that it wouldn't fit.
Next step is to multiply row 3 by (Bn-A^2)/ (2ABC-C^2n + BDn - A^2D - B^3)
1 A/n B/n | y/n
0 1 (Cn-AB)/(Bn-A^2) | (zn-Ay)/(Bn-A^2)
0 0 1 | (wAB+yAC+nwB-wA^2-yB^2)/(2ABC-nC^2+nBD-A^2D-B^3)
We're almost there!
Multiply row 3 by (AB-Cn)/(Bn-A^2) and add to row 2.
1 A/n B/n | y/n
0 1 0 | b
0 0 1 | c
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)
We've solved for b and c, but now we have to find a.
Take row 3 * (-B/n) and add to row 1.
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))
0 1 0 | b
0 0 1 | c
One more step!
Take row 2 * (-A/n) and add it to row 1.
Take a stab at finishing the rest.
That was a doozy!
Hope this helped!
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