5x-2y=1;3x+5y=13 eching

$$5 x - 2 y = 1 ; 3 x + 5 y = 13$$

x, y uchun yechish

$$x=1$$

$$y=2$$

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.

$$5x-2y=1,3x+5y=13$$

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

$$5x-2y=1$$

2y ni tenglamaning ikkala tarafiga qo'shish.

$$5x=2y+1$$

Ikki tarafini 5 ga bo‘ling.

$$x=\frac{1}{5}\left(2y+1\right)$$

\frac{1}{5} ni 2y+1 marotabaga ko'paytirish.

$$x=\frac{2}{5}y+\frac{1}{5}$$

\frac{2y+1}{5} ni x uchun boshqa tenglamada almashtirish, 3x+5y=13.

$$3\left(\frac{2}{5}y+\frac{1}{5}\right)+5y=13$$

3 ni \frac{2y+1}{5} marotabaga ko'paytirish.

$$\frac{6}{5}y+\frac{3}{5}+5y=13$$

\frac{6y}{5} ni 5y ga qo'shish.

$$\frac{31}{5}y+\frac{3}{5}=13$$

Tenglamaning ikkala tarafidan \frac{3}{5} ni ayirish.

$$\frac{31}{5}y=\frac{62}{5}$$

Tenglamaning ikki tarafini \frac{31}{5} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$y=2$$

2 ni y uchun x=\frac{2}{5}y+\frac{1}{5} da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=\frac{2}{5}\times 2+\frac{1}{5}$$

\frac{2}{5} ni 2 marotabaga ko'paytirish.

$$x=\frac{4+1}{5}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{5} ni \frac{4}{5} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=1$$

Tizim hal qilindi.

$$x=1,y=2$$

Matritsalardan foydalanish qadamlari

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

$$5x-2y=1,3x+5y=13$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}5&-2\\3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\13\end{matrix}\right)$$

\left(\begin{matrix}5&-2\\3&5\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}5&-2\\3&5\end{matrix}\right))\left(\begin{matrix}5&-2\\3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\3&5\end{matrix}\right))\left(\begin{matrix}1\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\3&5\end{matrix}\right))\left(\begin{matrix}1\\13\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\3&5\end{matrix}\right))\left(\begin{matrix}1\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5\times 5-\left(-2\times 3\right)}&-\frac{-2}{5\times 5-\left(-2\times 3\right)}\\-\frac{3}{5\times 5-\left(-2\times 3\right)}&\frac{5}{5\times 5-\left(-2\times 3\right)}\end{matrix}\right)\left(\begin{matrix}1\\13\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{31}&\frac{2}{31}\\-\frac{3}{31}&\frac{5}{31}\end{matrix}\right)\left(\begin{matrix}1\\13\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{31}+\frac{2}{31}\times 13\\-\frac{3}{31}+\frac{5}{31}\times 13\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\2\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=1,y=2$$

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.

$$5x-2y=1,3x+5y=13$$

5x va 3x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 3 ga va ikkinchining har bir tarafidagi barcha shartlarni 5 ga ko'paytiring.

$$3\times 5x+3\left(-2\right)y=3,5\times 3x+5\times 5y=5\times 13$$

Qisqartirish.

$$15x-6y=3,15x+25y=65$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 15x-6y=3 dan 15x+25y=65 ni ayirish.

$$15x-15x-6y-25y=3-65$$

15x ni -15x ga qo'shish. 15x va -15x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$-6y-25y=3-65$$

-6y ni -25y ga qo'shish.

$$-31y=3-65$$

3 ni -65 ga qo'shish.

$$-31y=-62$$

Ikki tarafini -31 ga bo‘ling.

$$y=2$$

2 ni y uchun 3x+5y=13 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$3x+5\times 2=13$$

5 ni 2 marotabaga ko'paytirish.

$$3x+10=13$$

Tenglamaning ikkala tarafidan 10 ni ayirish.

$$3x=3$$

Ikki tarafini 3 ga bo‘ling.

$$x=1$$

Tizim hal qilindi.

$$x=1,y=2$$