x-2y=-2;x+2y=10 eching

$$x - 2 y = - 2 ; x + 2 y = 10$$

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.

$$x-2y=-2,x+2y=10$$

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

$$x-2y=-2$$

2y ni tenglamaning ikkala tarafiga qo'shish.

$$x=2y-2$$

-2+2y ni x uchun boshqa tenglamada almashtirish, x+2y=10.

$$2y-2+2y=10$$

2y ni 2y ga qo'shish.

$$4y-2=10$$

2 ni tenglamaning ikkala tarafiga qo'shish.

$$4y=12$$

Ikki tarafini 4 ga bo‘ling.

$$y=3$$

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

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

2 ni 3 marotabaga ko'paytirish.

$$x=6-2$$

-2 ni 6 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.

$$x-2y=-2,x+2y=10$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}1&-2\\1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2\\10\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-2\\1&2\end{matrix}\right))\left(\begin{matrix}1&-2\\1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\1&2\end{matrix}\right))\left(\begin{matrix}-2\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\1&2\end{matrix}\right))\left(\begin{matrix}-2\\10\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\1&2\end{matrix}\right))\left(\begin{matrix}-2\\10\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2-\left(-2\right)}&-\frac{-2}{2-\left(-2\right)}\\-\frac{1}{2-\left(-2\right)}&\frac{1}{2-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}-2\\10\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{2}\\-\frac{1}{4}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}-2\\10\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\left(-2\right)+\frac{1}{2}\times 10\\-\frac{1}{4}\left(-2\right)+\frac{1}{4}\times 10\end{matrix}\right)$$

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.

$$x-2y=-2,x+2y=10$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali x-2y=-2 dan x+2y=10 ni ayirish.

$$x-x-2y-2y=-2-10$$

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

$$-2y-2y=-2-10$$

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

$$-4y=-2-10$$

-2 ni -10 ga qo'shish.

$$-4y=-12$$

Ikki tarafini -4 ga bo‘ling.

$$y=3$$

3 ni y uchun x+2y=10 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x+2\times 3=10$$

2 ni 3 marotabaga ko'paytirish.

$$x+6=10$$

Tenglamaning ikkala tarafidan 6 ni ayirish.

$$x=4$$

Tizim hal qilindi.

$$x=4,y=3$$