Question 30328: AX+BY=P
BX-AY=Q SOLVE FORX&Y
Answer by sdmmadam@yahoo.com(530)   (Show Source):
You can put this solution on YOUR website! AX+BY=P ----(1)multiplied by A
BX-AY=Q ----(2)multiplied by B
(1) is multiplied by A and (2) is multiplied by B
so that the coefficients of Y are made the same.
A^2X +(AB)Y = AP ----(3)
B^2X -(AB)Y = BQ ----(4)
(3) + (4) gives
(A^2+B^2)X = AP+BQ
Dividing by (A^2+B^2)
X = (AP+BQ)/(A^2+B^2) ----(*)
Putting (*) in (1)
AX+BY =P
A times[(AP+BQ)/(A^2+B^2)]+BY = P
Multiplying by (A^2+B^2)
A(AP+BQ) +B(A^2+B^2)Y = P(A^2+B^2)
B(A^2+B^2)Y = P(A^2+B^2)-A(AP+BQ)
B(A^2+B^2)Y = PA^2+PB^2-PA^2-ABQ
B(A^2+B^2)Y = (PA^2-PA^2)+PB^2-ABQ
B(A^2+B^2)Y = (PA^2-PA^2)+PB^2-ABQ
B(A^2+B^2)Y = 0+B(BP-AQ)
B(A^2+B^2)Y = B(BP-AQ)
Dividing by B(A^2+B^2)
Y = (BP-AQ)/(A^2+B^2)
Answer: X = (AP+BQ)/(A^2+B^2) and Y = (BP-AQ)/(A^2+B^2)
Note:Instead of substituting for X and getting Y,we may give a similar treatment of equalising coefficients and using subtractilon to get rid of X to get Y.
Verification:Since (1) was used to find Y substituting for X we shall
use (2) for verification.
BX-AY=Q ----(2)
Putting X = (AP+BQ)/(A^2+B^2) and Y = (BP-AQ)/(A^2+B^2)in this
LHS = B(AP+BQ)/(A^2+B^2) - A(BP-AQ)/(A^2+B^2)
=[1/(A^2+B^2)]multiplied by[B(AP+BQ)- A(BP-AQ)]
=[1/(A^2+B^2)]multiplied by(ABP+B^2Q-ABP+A^2Q]
=(ABP-ABP+A^2Q+B^2Q]/(A^2+B^2)
=[0+(A^2+B^2)Q]/(A^2+B^2)
=(A^2+B^2)Q]/(A^2+B^2)
=Q (cancelling (A^2+B^2)
=RHS
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