3x+9=6ytext{and}-2x-2y-12=0 eching

$$3 x + 9 = 6 y \text { and } - 2 x - 2 y - 12 = 0$$

x, y uchun yechish

$$x=-5$$

$$y=-1$$

O'rnini bosishdan foydalanish qoidalari

Birinchi tenglamani yeching. Ikkala tarafdan 6y ni ayirish.

$$3x+9-6y=0$$

Ikkala tarafdan 9 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$3x-6y=-9$$

Ikkinchi tenglamani yeching. 12 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$-2x-2y=12$$

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.

$$3x-6y=-9,-2x-2y=12$$

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

$$3x-6y=-9$$

6y ni tenglamaning ikkala tarafiga qo'shish.

$$3x=6y-9$$

Ikki tarafini 3 ga bo‘ling.

$$x=\frac{1}{3}\left(6y-9\right)$$

\frac{1}{3} ni 6y-9 marotabaga ko'paytirish.

$$x=2y-3$$

2y-3 ni x uchun boshqa tenglamada almashtirish, -2x-2y=12.

$$-2\left(2y-3\right)-2y=12$$

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

$$-4y+6-2y=12$$

-4y ni -2y ga qo'shish.

$$-6y+6=12$$

Tenglamaning ikkala tarafidan 6 ni ayirish.

$$-6y=6$$

Ikki tarafini -6 ga bo‘ling.

$$y=-1$$

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

$$x=2\left(-1\right)-3$$

2 ni -1 marotabaga ko'paytirish.

$$x=-2-3$$

-3 ni -2 ga qo'shish.

$$x=-5$$

Tizim hal qilindi.

$$x=-5,y=-1$$

Matritsalardan foydalanish qadamlari

Birinchi tenglamani yeching. Ikkala tarafdan 6y ni ayirish.

$$3x+9-6y=0$$

Ikkala tarafdan 9 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$3x-6y=-9$$

Ikkinchi tenglamani yeching. 12 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$-2x-2y=12$$

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

$$3x-6y=-9,-2x-2y=12$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}3&-6\\-2&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-9\\12\end{matrix}\right)$$

\left(\begin{matrix}3&-6\\-2&-2\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}3&-6\\-2&-2\end{matrix}\right))\left(\begin{matrix}3&-6\\-2&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-6\\-2&-2\end{matrix}\right))\left(\begin{matrix}-9\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-6\\-2&-2\end{matrix}\right))\left(\begin{matrix}-9\\12\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-6\\-2&-2\end{matrix}\right))\left(\begin{matrix}-9\\12\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{3\left(-2\right)-\left(-6\left(-2\right)\right)}&-\frac{-6}{3\left(-2\right)-\left(-6\left(-2\right)\right)}\\-\frac{-2}{3\left(-2\right)-\left(-6\left(-2\right)\right)}&\frac{3}{3\left(-2\right)-\left(-6\left(-2\right)\right)}\end{matrix}\right)\left(\begin{matrix}-9\\12\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}&-\frac{1}{3}\\-\frac{1}{9}&-\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}-9\\12\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}\left(-9\right)-\frac{1}{3}\times 12\\-\frac{1}{9}\left(-9\right)-\frac{1}{6}\times 12\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

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

x va y matritsa elementlarini chiqarib olish.

$$x=-5,y=-1$$

Chiqarib tashlashdan foydalanish qadamlari

Birinchi tenglamani yeching. Ikkala tarafdan 6y ni ayirish.

$$3x+9-6y=0$$

Ikkala tarafdan 9 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$3x-6y=-9$$

Ikkinchi tenglamani yeching. 12 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$-2x-2y=12$$

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.

$$3x-6y=-9,-2x-2y=12$$

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.

$$-2\times 3x-2\left(-6\right)y=-2\left(-9\right),3\left(-2\right)x+3\left(-2\right)y=3\times 12$$

Qisqartirish.

$$-6x+12y=18,-6x-6y=36$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali -6x+12y=18 dan -6x-6y=36 ni ayirish.

$$-6x+6x+12y+6y=18-36$$

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

$$12y+6y=18-36$$

12y ni 6y ga qo'shish.

$$18y=18-36$$

18 ni -36 ga qo'shish.

$$18y=-18$$

Ikki tarafini 18 ga bo‘ling.

$$y=-1$$

-1 ni y uchun -2x-2y=12 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$-2x-2\left(-1\right)=12$$

-2 ni -1 marotabaga ko'paytirish.

$$-2x+2=12$$

Tenglamaning ikkala tarafidan 2 ni ayirish.

$$-2x=10$$

Ikki tarafini -2 ga bo‘ling.

$$x=-5$$

Tizim hal qilindi.

$$x=-5,y=-1$$