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8x+2y=46,7x+3y=47
Almashtirishdan foydalanib tenglamalar juftligini yechish uchun, avval o'zgaruvchan qiymatlardan biri uchun tenglamani yeching. So'ngra ana shu o'zgaruvchan natijani boshqa tenglama bilan almashtiring.
8x+2y=46
Tenglamalardan birini tanlang va teng belgisining chap tomonidagi x ni izolyatsiyalash orqali x ni hisoblang.
8x=-2y+46
Tenglamaning ikkala tarafidan 2y ni ayirish.
x=\frac{1}{8}\left(-2y+46\right)
Ikki tarafini 8 ga bo‘ling.
x=-\frac{1}{4}y+\frac{23}{4}
\frac{1}{8} ni -2y+46 marotabaga ko'paytirish.
7\left(-\frac{1}{4}y+\frac{23}{4}\right)+3y=47
\frac{-y+23}{4} ni x uchun boshqa tenglamada almashtirish, 7x+3y=47.
-\frac{7}{4}y+\frac{161}{4}+3y=47
7 ni \frac{-y+23}{4} marotabaga ko'paytirish.
\frac{5}{4}y+\frac{161}{4}=47
-\frac{7y}{4} ni 3y ga qo'shish.
\frac{5}{4}y=\frac{27}{4}
Tenglamaning ikkala tarafidan \frac{161}{4} ni ayirish.
y=\frac{27}{5}
Tenglamaning ikki tarafini \frac{5}{4} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.
x=-\frac{1}{4}\times \frac{27}{5}+\frac{23}{4}
\frac{27}{5} ni y uchun x=-\frac{1}{4}y+\frac{23}{4} da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.
x=-\frac{27}{20}+\frac{23}{4}
Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali -\frac{1}{4} ni \frac{27}{5} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.
x=\frac{22}{5}
Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{23}{4} ni -\frac{27}{20} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.
x=\frac{22}{5},y=\frac{27}{5}
Tizim hal qilindi.
8x+2y=46,7x+3y=47
Tenglamalar standart shaklda ko'rsatilsin so'ng tenglamalar tizimini yechish uchun matritsalardan foydalanilsin.
\left(\begin{matrix}8&2\\7&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}46\\47\end{matrix}\right)
Tenglamalarni matritsa shaklida yozish.
inverse(\left(\begin{matrix}8&2\\7&3\end{matrix}\right))\left(\begin{matrix}8&2\\7&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&2\\7&3\end{matrix}\right))\left(\begin{matrix}46\\47\end{matrix}\right)
\left(\begin{matrix}8&2\\7&3\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.
\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&2\\7&3\end{matrix}\right))\left(\begin{matrix}46\\47\end{matrix}\right)
Matritsaning ko‘paytmasi va teskarisi o‘zaro teng matristsadir.
\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&2\\7&3\end{matrix}\right))\left(\begin{matrix}46\\47\end{matrix}\right)
Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8\times 3-2\times 7}&-\frac{2}{8\times 3-2\times 7}\\-\frac{7}{8\times 3-2\times 7}&\frac{8}{8\times 3-2\times 7}\end{matrix}\right)\left(\begin{matrix}46\\47\end{matrix}\right)
\left(\begin{matrix}a&b\\c&d\end{matrix}\right) 2\times 2 matrix uchun, teskari matritsa \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), shuning uchun matritsa tenglamasini matritsani ko‘paytirish masalasi sifatida qayta yozish mumkin.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{10}&-\frac{1}{5}\\-\frac{7}{10}&\frac{4}{5}\end{matrix}\right)\left(\begin{matrix}46\\47\end{matrix}\right)
Arifmetik hisobni amalga oshirish.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{10}\times 46-\frac{1}{5}\times 47\\-\frac{7}{10}\times 46+\frac{4}{5}\times 47\end{matrix}\right)
Matritsalarni ko'paytirish.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{22}{5}\\\frac{27}{5}\end{matrix}\right)
Arifmetik hisobni amalga oshirish.
x=\frac{22}{5},y=\frac{27}{5}
x va y matritsa elementlarini chiqarib olish.
8x+2y=46,7x+3y=47
Chiqarib tashlash bilan yechim hosil qilish uchun, o'zgartmalarning koeffitsienti ikkala tenglamada bir xil bo'lib o'zgaruvchan qiymat birining boshqasidan ayirilganda, bekor qilishi lozim.
7\times 8x+7\times 2y=7\times 46,8\times 7x+8\times 3y=8\times 47
8x va 7x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 7 ga va ikkinchining har bir tarafidagi barcha shartlarni 8 ga ko'paytiring.
56x+14y=322,56x+24y=376
Qisqartirish.
56x-56x+14y-24y=322-376
Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 56x+14y=322 dan 56x+24y=376 ni ayirish.
14y-24y=322-376
56x ni -56x ga qo'shish. 56x va -56x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.
-10y=322-376
14y ni -24y ga qo'shish.
-10y=-54
322 ni -376 ga qo'shish.
y=\frac{27}{5}
Ikki tarafini -10 ga bo‘ling.
7x+3\times \frac{27}{5}=47
\frac{27}{5} ni y uchun 7x+3y=47 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.
7x+\frac{81}{5}=47
3 ni \frac{27}{5} marotabaga ko'paytirish.
7x=\frac{154}{5}
Tenglamaning ikkala tarafidan \frac{81}{5} ni ayirish.
x=\frac{22}{5}
Ikki tarafini 7 ga bo‘ling.
x=\frac{22}{5},y=\frac{27}{5}
Tizim hal qilindi.