{l}{x+2y=60}{x-y=30} eching

$$\left\{ \begin{array} { l } { x + 2 y = 60 } \\ { x - y = 30 } \end{array} \right\}$$

x, y uchun yechish

$$x=40$$

$$y=10$$

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.

$$x+2y=60,x-y=30$$

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

$$x+2y=60$$

Tenglamaning ikkala tarafidan 2y ni ayirish.

$$x=-2y+60$$

-2y+60 ni x uchun boshqa tenglamada almashtirish, x-y=30.

$$-2y+60-y=30$$

-2y ni -y ga qo'shish.

$$-3y+60=30$$

Tenglamaning ikkala tarafidan 60 ni ayirish.

$$-3y=-30$$

Ikki tarafini -3 ga bo‘ling.

$$y=10$$

10 ni y uchun x=-2y+60 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=-2\times 10+60$$

-2 ni 10 marotabaga ko'paytirish.

$$x=-20+60$$

60 ni -20 ga qo'shish.

$$x=40$$

Tizim hal qilindi.

$$x=40,y=10$$

Matritsalardan foydalanish qadamlari

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

$$x+2y=60,x-y=30$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}1&2\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}60\\30\end{matrix}\right)$$

\left(\begin{matrix}1&2\\1&-1\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}1&2\\1&-1\end{matrix}\right))\left(\begin{matrix}1&2\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\1&-1\end{matrix}\right))\left(\begin{matrix}60\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\1&-1\end{matrix}\right))\left(\begin{matrix}60\\30\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\1&-1\end{matrix}\right))\left(\begin{matrix}60\\30\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-2}&-\frac{2}{-1-2}\\-\frac{1}{-1-2}&\frac{1}{-1-2}\end{matrix}\right)\left(\begin{matrix}60\\30\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&\frac{2}{3}\\\frac{1}{3}&-\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}60\\30\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 60+\frac{2}{3}\times 30\\\frac{1}{3}\times 60-\frac{1}{3}\times 30\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}40\\10\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=40,y=10$$

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.

$$x+2y=60,x-y=30$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali x+2y=60 dan x-y=30 ni ayirish.

$$x-x+2y+y=60-30$$

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

$$2y+y=60-30$$

2y ni y ga qo'shish.

$$3y=60-30$$

60 ni -30 ga qo'shish.

$$3y=30$$

Ikki tarafini 3 ga bo‘ling.

$$y=10$$

10 ni y uchun x-y=30 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x-10=30$$

10 ni tenglamaning ikkala tarafiga qo'shish.

$$x=40$$

Tizim hal qilindi.

$$x=40,y=10$$