4x-5y=2text{and}x+10y=41 eching

$$4 x - 5 y = 2 \text { and } x + 10 y = 41$$

x, y uchun yechish

$$x=5$$

$$y = \frac{18}{5} = 3\frac{3}{5} = 3,6$$

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.

$$4x-5y=2,x+10y=41$$

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

$$4x-5y=2$$

5y ni tenglamaning ikkala tarafiga qo'shish.

$$4x=5y+2$$

Ikki tarafini 4 ga bo‘ling.

$$x=\frac{1}{4}\left(5y+2\right)$$

\frac{1}{4} ni 5y+2 marotabaga ko'paytirish.

$$x=\frac{5}{4}y+\frac{1}{2}$$

\frac{5y}{4}+\frac{1}{2} ni x uchun boshqa tenglamada almashtirish, x+10y=41.

$$\frac{5}{4}y+\frac{1}{2}+10y=41$$

\frac{5y}{4} ni 10y ga qo'shish.

$$\frac{45}{4}y+\frac{1}{2}=41$$

Tenglamaning ikkala tarafidan \frac{1}{2} ni ayirish.

$$\frac{45}{4}y=\frac{81}{2}$$

Tenglamaning ikki tarafini \frac{45}{4} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$y=\frac{18}{5}$$

\frac{18}{5} ni y uchun x=\frac{5}{4}y+\frac{1}{2} da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=\frac{5}{4}\times \frac{18}{5}+\frac{1}{2}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{5}{4} ni \frac{18}{5} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$x=\frac{9+1}{2}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{2} ni \frac{9}{2} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=5$$

Tizim hal qilindi.

$$x=5,y=\frac{18}{5}$$

Matritsalardan foydalanish qadamlari

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

$$4x-5y=2,x+10y=41$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}4&-5\\1&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\41\end{matrix}\right)$$

\left(\begin{matrix}4&-5\\1&10\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}4&-5\\1&10\end{matrix}\right))\left(\begin{matrix}4&-5\\1&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\1&10\end{matrix}\right))\left(\begin{matrix}2\\41\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\1&10\end{matrix}\right))\left(\begin{matrix}2\\41\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\1&10\end{matrix}\right))\left(\begin{matrix}2\\41\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{4\times 10-\left(-5\right)}&-\frac{-5}{4\times 10-\left(-5\right)}\\-\frac{1}{4\times 10-\left(-5\right)}&\frac{4}{4\times 10-\left(-5\right)}\end{matrix}\right)\left(\begin{matrix}2\\41\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{9}&\frac{1}{9}\\-\frac{1}{45}&\frac{4}{45}\end{matrix}\right)\left(\begin{matrix}2\\41\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{9}\times 2+\frac{1}{9}\times 41\\-\frac{1}{45}\times 2+\frac{4}{45}\times 41\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\\frac{18}{5}\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=5,y=\frac{18}{5}$$

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.

$$4x-5y=2,x+10y=41$$

4x va x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 1 ga va ikkinchining har bir tarafidagi barcha shartlarni 4 ga ko'paytiring.

$$4x-5y=2,4x+4\times 10y=4\times 41$$

Qisqartirish.

$$4x-5y=2,4x+40y=164$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 4x-5y=2 dan 4x+40y=164 ni ayirish.

$$4x-4x-5y-40y=2-164$$

4x ni -4x ga qo'shish. 4x va -4x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$-5y-40y=2-164$$

-5y ni -40y ga qo'shish.

$$-45y=2-164$$

2 ni -164 ga qo'shish.

$$-45y=-162$$

Ikki tarafini -45 ga bo‘ling.

$$y=\frac{18}{5}$$

\frac{18}{5} ni y uchun x+10y=41 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x+10\times \frac{18}{5}=41$$

10 ni \frac{18}{5} marotabaga ko'paytirish.

$$x+36=41$$

Tenglamaning ikkala tarafidan 36 ni ayirish.

$$x=5$$

Tizim hal qilindi.

$$x=5,y=\frac{18}{5}$$