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Miscellaneous Exercise 4(A) · Q70

Q.Find the value of k if the following equations are consistent: (k−2)x+(k−1)y=17, (k−1)x+(k−2)y=18, x+y=5(k-2)x+(k-1)y=17,\ (k-1)x+(k-2)y=18,\ x+y=5

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D=∣k−2k−1−17k−1k−2−1811−5∣D=\begin{vmatrix}k-2&k-1&-17\\k-1&k-2&-18\\1&1&-5\end{vmatrix}. Expand along row 3: C31=∣k−1−17k−2−18∣=−18(k−1)+17(k−2)=−k−16C_{31}=\begin{vmatrix}k-1&-17\\k-2&-18\end{vmatrix}=-18(k-1)+17(k-2)=-k-16; C32=−∣k−2−17k−1−18∣=−[−18(k−2)+17(k−1)]=k−19C_{32}=-\begin{vmatrix}k-2&-17\\k-1&-18\end{vmatrix}=-[-18(k-2)+17(k-1)]=k-19; C33=∣k−2k−1k−1k−2∣=(k−2)2−(k−1)2=3−2kC_{33}=\begin{vmatrix}k-2&k-1\\k-1&k-2\end{vmatrix}=(k-2)^2-(k-1)^2=3-2k. …

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