x-y=5text{and}5y-4x=7 eching

$$x - y = 5 \text { and } 5 y - 4 x = 7$$

x, y uchun yechish

$$x=32$$

$$y=27$$

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.

$$x-y=5,-4x+5y=7$$

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

$$x-y=5$$

y ni tenglamaning ikkala tarafiga qo'shish.

$$x=y+5$$

y+5 ni x uchun boshqa tenglamada almashtirish, -4x+5y=7.

$$-4\left(y+5\right)+5y=7$$

-4 ni y+5 marotabaga ko'paytirish.

$$-4y-20+5y=7$$

-4y ni 5y ga qo'shish.

$$y-20=7$$

20 ni tenglamaning ikkala tarafiga qo'shish.

$$y=27$$

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

$$x=27+5$$

5 ni 27 ga qo'shish.

$$x=32$$

Tizim hal qilindi.

$$x=32,y=27$$

Matritsalardan foydalanish qadamlari

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

$$x-y=5,-4x+5y=7$$

Tenglamalarni matritsa shaklida yozish.

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

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

$$inverse(\left(\begin{matrix}1&-1\\-4&5\end{matrix}\right))\left(\begin{matrix}1&-1\\-4&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\-4&5\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\-4&5\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\-4&5\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

\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.

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

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5&1\\4&1\end{matrix}\right)\left(\begin{matrix}5\\7\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\times 5+7\\4\times 5+7\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}32\\27\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=32,y=27$$

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.

$$x-y=5,-4x+5y=7$$

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

$$-4x-4\left(-1\right)y=-4\times 5,-4x+5y=7$$

Qisqartirish.

$$-4x+4y=-20,-4x+5y=7$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali -4x+4y=-20 dan -4x+5y=7 ni ayirish.

$$-4x+4x+4y-5y=-20-7$$

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

$$4y-5y=-20-7$$

4y ni -5y ga qo'shish.

$$-y=-20-7$$

-20 ni -7 ga qo'shish.

$$-y=-27$$

Ikki tarafini -1 ga bo‘ling.

$$y=27$$

27 ni y uchun -4x+5y=7 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$-4x+5\times 27=7$$

5 ni 27 marotabaga ko'paytirish.

$$-4x+135=7$$

Tenglamaning ikkala tarafidan 135 ni ayirish.

$$-4x=-128$$

Ikki tarafini -4 ga bo‘ling.

$$x=32$$

Tizim hal qilindi.

$$x=32,y=27$$