{l}{8x+2y=46}{7x+3y=47} eching

$$\left\{ \begin{array} { l } { 8 x + 2 y = 46 } \\ { 7 x + 3 y = 47 } \end{array} \right.$$

x, y uchun yechish

$$x = \frac{22}{5} = 4\frac{2}{5} = 4,4$$

$$y = \frac{27}{5} = 5\frac{2}{5} = 5,4$$

O'rnini bosishdan foydalanish qoidalari

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.

$$8x+2y=46,7x+3y=47$$

Tenglamalardan birini tanlang va teng belgisining chap tomonidagi x ni izolyatsiyalash orqali x ni hisoblang.

$$8x+2y=46$$

Tenglamaning ikkala tarafidan 2y ni ayirish.

$$8x=-2y+46$$

Ikki tarafini 8 ga bo‘ling.

$$x=\frac{1}{8}\left(-2y+46\right)$$

\frac{1}{8} ni -2y+46 marotabaga ko'paytirish.

$$x=-\frac{1}{4}y+\frac{23}{4}$$

\frac{-y+23}{4} ni x uchun boshqa tenglamada almashtirish, 7x+3y=47.

$$7\left(-\frac{1}{4}y+\frac{23}{4}\right)+3y=47$$

7 ni \frac{-y+23}{4} marotabaga ko'paytirish.

$$-\frac{7}{4}y+\frac{161}{4}+3y=47$$

-\frac{7y}{4} ni 3y ga qo'shish.

$$\frac{5}{4}y+\frac{161}{4}=47$$

Tenglamaning ikkala tarafidan \frac{161}{4} ni ayirish.

$$\frac{5}{4}y=\frac{27}{4}$$

Tenglamaning ikki tarafini \frac{5}{4} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$y=\frac{27}{5}$$

\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.

$$x=-\frac{1}{4}\times \frac{27}{5}+\frac{23}{4}$$

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.

$$x=-\frac{27}{20}+\frac{23}{4}$$

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.

$$x=\frac{22}{5}$$

Tizim hal qilindi.

$$x=\frac{22}{5},y=\frac{27}{5}$$

Matritsalardan foydalanish qadamlari

Tenglamalar standart shaklda ko'rsatilsin so'ng tenglamalar tizimini yechish uchun matritsalardan foydalanilsin.

$$8x+2y=46,7x+3y=47$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}8&2\\7&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}46\\47\end{matrix}\right)$$

\left(\begin{matrix}8&2\\7&3\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}8&2\\7&3\end{matrix}\right))\left(\begin{matrix}8&2\\7&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&2\\7&3\end{matrix}\right))\left(\begin{matrix}46\\47\end{matrix}\right)$$

Matritsaning ko‘paytmasi va teskarisi o‘zaro teng matristsadir.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&2\\7&3\end{matrix}\right))\left(\begin{matrix}46\\47\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&2\\7&3\end{matrix}\right))\left(\begin{matrix}46\\47\end{matrix}\right)$$

2\times 2 matritsasi uchun \left(\begin{matrix}a&b\\c&d\end{matrix}\right), inversiyali matritsa \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), shu bois matritsa tenglamasini matritsaga ko‘paytirish muammosi sifatida qayta yozilishi mumkin.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8\times 3-2\times 7}&-\frac{2}{8\times 3-2\times 7}\\-\frac{7}{8\times 3-2\times 7}&\frac{8}{8\times 3-2\times 7}\end{matrix}\right)\left(\begin{matrix}46\\47\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{10}&-\frac{1}{5}\\-\frac{7}{10}&\frac{4}{5}\end{matrix}\right)\left(\begin{matrix}46\\47\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{10}\times 46-\frac{1}{5}\times 47\\-\frac{7}{10}\times 46+\frac{4}{5}\times 47\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{22}{5}\\\frac{27}{5}\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=\frac{22}{5},y=\frac{27}{5}$$

Chiqarib tashlashdan foydalanish qadamlari

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.

$$8x+2y=46,7x+3y=47$$

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.

$$7\times 8x+7\times 2y=7\times 46,8\times 7x+8\times 3y=8\times 47$$

Qisqartirish.

$$56x+14y=322,56x+24y=376$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 56x+14y=322 dan 56x+24y=376 ni ayirish.

$$56x-56x+14y-24y=322-376$$

56x ni -56x ga qo'shish. 56x va -56x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$14y-24y=322-376$$

14y ni -24y ga qo'shish.

$$-10y=322-376$$

322 ni -376 ga qo'shish.

$$-10y=-54$$

Ikki tarafini -10 ga bo‘ling.

$$y=\frac{27}{5}$$

\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.

$$7x+3\times \frac{27}{5}=47$$

3 ni \frac{27}{5} marotabaga ko'paytirish.

$$7x+\frac{81}{5}=47$$

Tenglamaning ikkala tarafidan \frac{81}{5} ni ayirish.

$$7x=\frac{154}{5}$$

Ikki tarafini 7 ga bo‘ling.

$$x=\frac{22}{5}$$

Tizim hal qilindi.

$$x=\frac{22}{5},y=\frac{27}{5}$$