49x-57y=172;quad57x-49y=252 eching

$$49 x - 57 y = 172 ; \quad 57 x - 49 y = 252$$

x, y uchun yechish

$$x=7$$

$$y=3$$

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.

$$49x-57y=172,57x-49y=252$$

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

$$49x-57y=172$$

57y ni tenglamaning ikkala tarafiga qo'shish.

$$49x=57y+172$$

Ikki tarafini 49 ga bo‘ling.

$$x=\frac{1}{49}\left(57y+172\right)$$

\frac{1}{49} ni 57y+172 marotabaga ko'paytirish.

$$x=\frac{57}{49}y+\frac{172}{49}$$

\frac{57y+172}{49} ni x uchun boshqa tenglamada almashtirish, 57x-49y=252.

$$57\left(\frac{57}{49}y+\frac{172}{49}\right)-49y=252$$

57 ni \frac{57y+172}{49} marotabaga ko'paytirish.

$$\frac{3249}{49}y+\frac{9804}{49}-49y=252$$

\frac{3249y}{49} ni -49y ga qo'shish.

$$\frac{848}{49}y+\frac{9804}{49}=252$$

Tenglamaning ikkala tarafidan \frac{9804}{49} ni ayirish.

$$\frac{848}{49}y=\frac{2544}{49}$$

Tenglamaning ikki tarafini \frac{848}{49} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$y=3$$

3 ni y uchun x=\frac{57}{49}y+\frac{172}{49} da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=\frac{57}{49}\times 3+\frac{172}{49}$$

\frac{57}{49} ni 3 marotabaga ko'paytirish.

$$x=\frac{171+172}{49}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{172}{49} ni \frac{171}{49} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=7$$

Tizim hal qilindi.

$$x=7,y=3$$

Matritsalardan foydalanish qadamlari

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

$$49x-57y=172,57x-49y=252$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}49&-57\\57&-49\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}172\\252\end{matrix}\right)$$

\left(\begin{matrix}49&-57\\57&-49\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}49&-57\\57&-49\end{matrix}\right))\left(\begin{matrix}49&-57\\57&-49\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}49&-57\\57&-49\end{matrix}\right))\left(\begin{matrix}172\\252\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}49&-57\\57&-49\end{matrix}\right))\left(\begin{matrix}172\\252\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}49&-57\\57&-49\end{matrix}\right))\left(\begin{matrix}172\\252\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{49}{49\left(-49\right)-\left(-57\times 57\right)}&-\frac{-57}{49\left(-49\right)-\left(-57\times 57\right)}\\-\frac{57}{49\left(-49\right)-\left(-57\times 57\right)}&\frac{49}{49\left(-49\right)-\left(-57\times 57\right)}\end{matrix}\right)\left(\begin{matrix}172\\252\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{49}{848}&\frac{57}{848}\\-\frac{57}{848}&\frac{49}{848}\end{matrix}\right)\left(\begin{matrix}172\\252\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{49}{848}\times 172+\frac{57}{848}\times 252\\-\frac{57}{848}\times 172+\frac{49}{848}\times 252\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\3\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=7,y=3$$

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.

$$49x-57y=172,57x-49y=252$$

49x va 57x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 57 ga va ikkinchining har bir tarafidagi barcha shartlarni 49 ga ko'paytiring.

$$57\times 49x+57\left(-57\right)y=57\times 172,49\times 57x+49\left(-49\right)y=49\times 252$$

Qisqartirish.

$$2793x-3249y=9804,2793x-2401y=12348$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 2793x-3249y=9804 dan 2793x-2401y=12348 ni ayirish.

$$2793x-2793x-3249y+2401y=9804-12348$$

2793x ni -2793x ga qo'shish. 2793x va -2793x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$-3249y+2401y=9804-12348$$

-3249y ni 2401y ga qo'shish.

$$-848y=9804-12348$$

9804 ni -12348 ga qo'shish.

$$-848y=-2544$$

Ikki tarafini -848 ga bo‘ling.

$$y=3$$

3 ni y uchun 57x-49y=252 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$57x-49\times 3=252$$

-49 ni 3 marotabaga ko'paytirish.

$$57x-147=252$$

147 ni tenglamaning ikkala tarafiga qo'shish.

$$57x=399$$

Ikki tarafini 57 ga bo‘ling.

$$x=7$$

Tizim hal qilindi.

$$x=7,y=3$$