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

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

x, y uchun yechish

$$x=5$$

$$y=-5$$

O'rnini bosishdan foydalanish qoidalari

Birinchi tenglamani yeching. y ni ikki tarafga qo’shing.

$$x+y=0$$

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.

$$x+y=0,2x+y=5$$

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

$$x+y=0$$

Tenglamaning ikkala tarafidan y ni ayirish.

$$x=-y$$

-y ni x uchun boshqa tenglamada almashtirish, 2x+y=5.

$$2\left(-1\right)y+y=5$$

2 ni -y marotabaga ko'paytirish.

$$-2y+y=5$$

-2y ni y ga qo'shish.

$$-y=5$$

Ikki tarafini -1 ga bo‘ling.

$$y=-5$$

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

$$x=-\left(-5\right)$$

-1 ni -5 marotabaga ko'paytirish.

$$x=5$$

Tizim hal qilindi.

$$x=5,y=-5$$

Matritsalardan foydalanish qadamlari

Birinchi tenglamani yeching. y ni ikki tarafga qo’shing.

$$x+y=0$$

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

$$x+y=0,2x+y=5$$

Tenglamalarni matritsa shaklida yozish.

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

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

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

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

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

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

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

\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.

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

Arifmetik hisobni amalga oshirish.

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

Matritsalarni ko'paytirish.

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

x va y matritsa elementlarini chiqarib olish.

$$x=5,y=-5$$

Chiqarib tashlashdan foydalanish qadamlari

Birinchi tenglamani yeching. y ni ikki tarafga qo’shing.

$$x+y=0$$

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.

$$x+y=0,2x+y=5$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali x+y=0 dan 2x+y=5 ni ayirish.

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

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

$$x-2x=-5$$

x ni -2x ga qo'shish.

$$-x=-5$$

Ikki tarafini -1 ga bo‘ling.

$$x=5$$

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

$$2\times 5+y=5$$

2 ni 5 marotabaga ko'paytirish.

$$10+y=5$$

Tenglamaning ikkala tarafidan 10 ni ayirish.

$$y=-5$$

Tizim hal qilindi.

$$x=5,y=-5$$