Using matrix method, solve the simultaneous equations {x-3y=3
5x-9y=11
Axˉ=bˉA\bar x=\bar bAxˉ=bˉ
A=(1−35−9),bˉ=(311)A=\begin{pmatrix} 1 & -3 \\ 5 & -9 \end{pmatrix},\bar b=\begin{pmatrix} 3 \\ 11 \end{pmatrix}A=(15−3−9),bˉ=(311) .
det(A)=−9+15=6.det(A)=-9+15=6.det(A)=−9+15=6.
A−1=16(−93−51)A^{-1}=\frac{1}{6}\begin{pmatrix} -9 & 3 \\ -5 & 1 \end{pmatrix}A−1=61(−9−531)
xˉ=(xy)=A−1bˉ=16(−93−51)(311)=16(6−4)=(1−23)\bar x=\begin{pmatrix} x \\ y \end{pmatrix}=A^{-1}\bar b=\frac{1}{6}\begin{pmatrix} -9& 3 \\ -5 & 1 \end{pmatrix}\begin{pmatrix} 3 \\ 11 \end{pmatrix}=\frac{1}{6}\begin{pmatrix} 6 \\ -4 \end{pmatrix}=\begin{pmatrix} 1 \\ -\frac{2}{3} \end{pmatrix}xˉ=(xy)=A−1bˉ=61(−9−531)(311)=61(6−4)=(1−32)
Thus, x=1,y=−23.x=1,y=-\frac{2}{3}.x=1,y=−32.
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