{l}{2x+y=3}{x+y=5} eching

$$\left\{ \begin{array} { l } { 2 x + y = 3 } \\ { x + y = 5 } \end{array} \right\}$$

x, y uchun yechish

$$x=-2$$

$$y=7$$

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.

$$2x+y=3,x+y=5$$

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

$$2x+y=3$$

Tenglamaning ikkala tarafidan y ni ayirish.

$$2x=-y+3$$

Ikki tarafini 2 ga bo‘ling.

$$x=\frac{1}{2}\left(-y+3\right)$$

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

$$x=-\frac{1}{2}y+\frac{3}{2}$$

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

$$-\frac{1}{2}y+\frac{3}{2}+y=5$$

-\frac{y}{2} ni y ga qo'shish.

$$\frac{1}{2}y+\frac{3}{2}=5$$

Tenglamaning ikkala tarafidan \frac{3}{2} ni ayirish.

$$\frac{1}{2}y=\frac{7}{2}$$

Ikkala tarafini 2 ga ko‘paytiring.

$$y=7$$

7 ni y uchun x=-\frac{1}{2}y+\frac{3}{2} da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=-\frac{1}{2}\times 7+\frac{3}{2}$$

-\frac{1}{2} ni 7 marotabaga ko'paytirish.

$$x=\frac{-7+3}{2}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{3}{2} ni -\frac{7}{2} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=-2$$

Tizim hal qilindi.

$$x=-2,y=7$$

Matritsalardan foydalanish qadamlari

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

$$2x+y=3,x+y=5$$

Tenglamalarni matritsa shaklida yozish.

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

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

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

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

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

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

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

\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.

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

Arifmetik hisobni amalga oshirish.

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

Matritsalarni ko'paytirish.

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

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2\\7\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=-2,y=7$$

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.

$$2x+y=3,x+y=5$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 2x+y=3 dan x+y=5 ni ayirish.

$$2x-x+y-y=3-5$$

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

$$2x-x=3-5$$

2x ni -x ga qo'shish.

$$x=3-5$$

3 ni -5 ga qo'shish.

$$x=-2$$

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

$$-2+y=5$$

2 ni tenglamaning ikkala tarafiga qo'shish.

$$y=7$$

Tizim hal qilindi.

$$x=-2,y=7$$