x+2y=-1;2x-3y=12 eching

$$x + 2 y = - 1 ; 2 x - 3 y = 12$$

x, y uchun yechish

$$x=3$$

$$y=-2$$

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.

$$x+2y=-1,2x-3y=12$$

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

$$x+2y=-1$$

Tenglamaning ikkala tarafidan 2y ni ayirish.

$$x=-2y-1$$

-2y-1 ni x uchun boshqa tenglamada almashtirish, 2x-3y=12.

$$2\left(-2y-1\right)-3y=12$$

2 ni -2y-1 marotabaga ko'paytirish.

$$-4y-2-3y=12$$

-4y ni -3y ga qo'shish.

$$-7y-2=12$$

2 ni tenglamaning ikkala tarafiga qo'shish.

$$-7y=14$$

Ikki tarafini -7 ga bo‘ling.

$$y=-2$$

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

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

-2 ni -2 marotabaga ko'paytirish.

$$x=4-1$$

-1 ni 4 ga qo'shish.

$$x=3$$

Tizim hal qilindi.

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

Matritsalardan foydalanish qadamlari

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

$$x+2y=-1,2x-3y=12$$

Tenglamalarni matritsa shaklida yozish.

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

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

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

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

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

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

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

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{-3-2\times 2}&-\frac{2}{-3-2\times 2}\\-\frac{2}{-3-2\times 2}&\frac{1}{-3-2\times 2}\end{matrix}\right)\left(\begin{matrix}-1\\12\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{7}&\frac{2}{7}\\\frac{2}{7}&-\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}-1\\12\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{7}\left(-1\right)+\frac{2}{7}\times 12\\\frac{2}{7}\left(-1\right)-\frac{1}{7}\times 12\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

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

x va y matritsa elementlarini chiqarib olish.

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

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.

$$x+2y=-1,2x-3y=12$$

x va 2x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 2 ga va ikkinchining har bir tarafidagi barcha shartlarni 1 ga ko'paytiring.

$$2x+2\times 2y=2\left(-1\right),2x-3y=12$$

Qisqartirish.

$$2x+4y=-2,2x-3y=12$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 2x+4y=-2 dan 2x-3y=12 ni ayirish.

$$2x-2x+4y+3y=-2-12$$

2x ni -2x ga qo'shish. 2x va -2x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$4y+3y=-2-12$$

4y ni 3y ga qo'shish.

$$7y=-2-12$$

-2 ni -12 ga qo'shish.

$$7y=-14$$

Ikki tarafini 7 ga bo‘ling.

$$y=-2$$

-2 ni y uchun 2x-3y=12 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

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

-3 ni -2 marotabaga ko'paytirish.

$$2x+6=12$$

Tenglamaning ikkala tarafidan 6 ni ayirish.

$$2x=6$$

Ikki tarafini 2 ga bo‘ling.

$$x=3$$

Tizim hal qilindi.

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