3x-y=9text{art}2x+3y=17 eching

$$3 x - y = 9 \text { art } 2 x + 3 y = 17$$

x, y uchun yechish

$$x=4$$

$$y=3$$

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.

$$3x-y=9,2x+3y=17$$

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

$$3x-y=9$$

y ni tenglamaning ikkala tarafiga qo'shish.

$$3x=y+9$$

Ikki tarafini 3 ga bo‘ling.

$$x=\frac{1}{3}\left(y+9\right)$$

\frac{1}{3} ni y+9 marotabaga ko'paytirish.

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

\frac{y}{3}+3 ni x uchun boshqa tenglamada almashtirish, 2x+3y=17.

$$2\left(\frac{1}{3}y+3\right)+3y=17$$

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

$$\frac{2}{3}y+6+3y=17$$

\frac{2y}{3} ni 3y ga qo'shish.

$$\frac{11}{3}y+6=17$$

Tenglamaning ikkala tarafidan 6 ni ayirish.

$$\frac{11}{3}y=11$$

Tenglamaning ikki tarafini \frac{11}{3} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$y=3$$

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

$$x=\frac{1}{3}\times 3+3$$

\frac{1}{3} ni 3 marotabaga ko'paytirish.

$$x=1+3$$

3 ni 1 ga qo'shish.

$$x=4$$

Tizim hal qilindi.

$$x=4,y=3$$

Matritsalardan foydalanish qadamlari

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

$$3x-y=9,2x+3y=17$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}3&-1\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\17\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&-1\\2&3\end{matrix}\right))\left(\begin{matrix}3&-1\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\2&3\end{matrix}\right))\left(\begin{matrix}9\\17\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\2&3\end{matrix}\right))\left(\begin{matrix}9\\17\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\2&3\end{matrix}\right))\left(\begin{matrix}9\\17\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3\times 3-\left(-2\right)}&-\frac{-1}{3\times 3-\left(-2\right)}\\-\frac{2}{3\times 3-\left(-2\right)}&\frac{3}{3\times 3-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}9\\17\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{11}&\frac{1}{11}\\-\frac{2}{11}&\frac{3}{11}\end{matrix}\right)\left(\begin{matrix}9\\17\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{11}\times 9+\frac{1}{11}\times 17\\-\frac{2}{11}\times 9+\frac{3}{11}\times 17\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\3\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=4,y=3$$

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.

$$3x-y=9,2x+3y=17$$

3x va 2x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 2 ga va ikkinchining har bir tarafidagi barcha shartlarni 3 ga ko'paytiring.

$$2\times 3x+2\left(-1\right)y=2\times 9,3\times 2x+3\times 3y=3\times 17$$

Qisqartirish.

$$6x-2y=18,6x+9y=51$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 6x-2y=18 dan 6x+9y=51 ni ayirish.

$$6x-6x-2y-9y=18-51$$

6x ni -6x ga qo'shish. 6x va -6x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$-2y-9y=18-51$$

-2y ni -9y ga qo'shish.

$$-11y=18-51$$

18 ni -51 ga qo'shish.

$$-11y=-33$$

Ikki tarafini -11 ga bo‘ling.

$$y=3$$

3 ni y uchun 2x+3y=17 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$2x+3\times 3=17$$

3 ni 3 marotabaga ko'paytirish.

$$2x+9=17$$

Tenglamaning ikkala tarafidan 9 ni ayirish.

$$2x=8$$

Ikki tarafini 2 ga bo‘ling.

$$x=4$$

Tizim hal qilindi.

$$x=4,y=3$$