7.4-bo'lim : Kengaytirilgan matritsa haqida ko'proq
3. Quyidagi tenglamalar sistemasi uchun tizimni kengaytirilgan matritsaga aylantiring va tizimning yechimini aniqlash yoki tizimning nomuvofiq yoki bog'liqligini aniqlash uchun to'ldirilgan matritsa usullaridan foydalaning.
\[\begin{align*}3x + 9y & = - 6\\ - 4x - 12y & = 8\end{align*}\]Yechim
Birinchi qadam tenglamalar sistemasi uchun kengaytirilgan matritsani yozishdir.
\[\left[ {\begin{array}{rr|r}3&9&{ - 6}\\{ - 4}&{ - 12}&8\end{array}} \right]\]2-qadam
Yuqori chap burchakdagi raqamni bitta qilishimiz kerak. Bunday holda, biz yuqori qatorni 3 ga bo'lish orqali buni tezda amalga oshirishimiz mumkin.
\[\left[ {\begin{array}{rr|r}3&9&{ - 6}\\{ - 4}&{ - 12}&8\end{array}} \right]\,\,\,\,\,\,\,\,\,\,\,\begin{array}{*{20}{c}}{\frac{1}{3}{R_{\,1}}}\\ \to \end{array}\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ {\begin{array}{rr|r}1&3&{ - 2}\\{ - 4}&{ - 12}&8\end{array}} \right]\]3-qadam
Keyinchalik, 1 dan pastdagi -4 ni nolga aylantirishimiz kerak va biz buni quyidagi elementar qator operatsiyasi bilan amalga oshirishimiz mumkin.
\[\left[ {\begin{array}{rr|r}1&3&{ - 2}\\{ - 4}&{ - 12}&8\end{array}} \right]\,\,\,\,\,\,\,\,\,\,\,\begin{array}{*{20}{c}}{{R_{\,2}} + 4{R_{\,1}} \to {R_{\,2}}}\\ \to \end{array}\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ {\begin{array}{rr|r}1&3&{ - 2}\\0&0&0\end{array}} \right]\]4-qadam
Biz barcha nollarning pastki qatorini ko'rganimizda, tizim agar ekanligini bilamiz qaram. Biz yuqori qatorni tenglamaga aylantira olamiz va \(x\) ni quyidagicha yechishimiz mumkin,
\[x + 3y = - 2\hspace{0.25in} \to \hspace{0.25in}x = - 3y - 2\]Bundan biz yechimni quyidagicha yozishimiz mumkin:
\[\begin{array}{l}{x = - 3t - 2}\\{y = t}\end{array}\hspace{0.25in}t{\mbox{ is any number}}\]