{l}{x+2y=7}{3x-2y=-3} eching

$$\left\{ \begin{array} { l } { x + 2 y = 7 } \\ { 3 x - 2 y = - 3 } \end{array} \right\}$$

x, y uchun yechish

$$x=1$$

$$y=3$$

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.

$$x+2y=7,3x-2y=-3$$

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

$$x+2y=7$$

Tenglamaning ikkala tarafidan 2y ni ayirish.

$$x=-2y+7$$

-2y+7 ni x uchun boshqa tenglamada almashtirish, 3x-2y=-3.

$$3\left(-2y+7\right)-2y=-3$$

3 ni -2y+7 marotabaga ko'paytirish.

$$-6y+21-2y=-3$$

-6y ni -2y ga qo'shish.

$$-8y+21=-3$$

Tenglamaning ikkala tarafidan 21 ni ayirish.

$$-8y=-24$$

Ikki tarafini -8 ga bo‘ling.

$$y=3$$

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

$$x=-2\times 3+7$$

-2 ni 3 marotabaga ko'paytirish.

$$x=-6+7$$

7 ni -6 ga qo'shish.

$$x=1$$

Tizim hal qilindi.

$$x=1,y=3$$

Matritsalardan foydalanish qadamlari

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

$$x+2y=7,3x-2y=-3$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}1&2\\3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\-3\end{matrix}\right)$$

\left(\begin{matrix}1&2\\3&-2\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}1&2\\3&-2\end{matrix}\right))\left(\begin{matrix}1&2\\3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\3&-2\end{matrix}\right))\left(\begin{matrix}7\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\3&-2\end{matrix}\right))\left(\begin{matrix}7\\-3\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\3&-2\end{matrix}\right))\left(\begin{matrix}7\\-3\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-2\times 3}&-\frac{2}{-2-2\times 3}\\-\frac{3}{-2-2\times 3}&\frac{1}{-2-2\times 3}\end{matrix}\right)\left(\begin{matrix}7\\-3\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&\frac{1}{4}\\\frac{3}{8}&-\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}7\\-3\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 7+\frac{1}{4}\left(-3\right)\\\frac{3}{8}\times 7-\frac{1}{8}\left(-3\right)\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

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

x va y matritsa elementlarini chiqarib olish.

$$x=1,y=3$$

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.

$$x+2y=7,3x-2y=-3$$

x va 3x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 3 ga va ikkinchining har bir tarafidagi barcha shartlarni 1 ga ko'paytiring.

$$3x+3\times 2y=3\times 7,3x-2y=-3$$

Qisqartirish.

$$3x+6y=21,3x-2y=-3$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 3x+6y=21 dan 3x-2y=-3 ni ayirish.

$$3x-3x+6y+2y=21+3$$

3x ni -3x ga qo'shish. 3x va -3x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$6y+2y=21+3$$

6y ni 2y ga qo'shish.

$$8y=21+3$$

21 ni 3 ga qo'shish.

$$8y=24$$

Ikki tarafini 8 ga bo‘ling.

$$y=3$$

3 ni y uchun 3x-2y=-3 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$3x-2\times 3=-3$$

-2 ni 3 marotabaga ko'paytirish.

$$3x-6=-3$$

6 ni tenglamaning ikkala tarafiga qo'shish.

$$3x=3$$

Ikki tarafini 3 ga bo‘ling.

$$x=1$$

Tizim hal qilindi.

$$x=1,y=3$$