{l}{y=3x}{x+y=16} eching

$$\left\{ \begin{array} { l } { y = 3 x } \\ { x + y = 16 } \end{array} \right\}$$

y, x uchun yechish

$$x=4$$

$$y=12$$

O'rnini bosishdan foydalanish qoidalari

Birinchi tenglamani yeching. Ikkala tarafdan 3x ni ayirish.

$$y-3x=0$$

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.

$$y-3x=0,y+x=16$$

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

$$y-3x=0$$

3x ni tenglamaning ikkala tarafiga qo'shish.

$$y=3x$$

3x ni y uchun boshqa tenglamada almashtirish, y+x=16.

$$3x+x=16$$

3x ni x ga qo'shish.

$$4x=16$$

Ikki tarafini 4 ga bo‘ling.

$$x=4$$

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

$$y=3\times 4$$

3 ni 4 marotabaga ko'paytirish.

$$y=12$$

Tizim hal qilindi.

$$y=12,x=4$$

Matritsalardan foydalanish qadamlari

Birinchi tenglamani yeching. Ikkala tarafdan 3x ni ayirish.

$$y-3x=0$$

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

$$y-3x=0,y+x=16$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}1&-3\\1&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\16\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-3\\1&1\end{matrix}\right))\left(\begin{matrix}1&-3\\1&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\1&1\end{matrix}\right))\left(\begin{matrix}0\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\1&1\end{matrix}\right))\left(\begin{matrix}0\\16\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\1&1\end{matrix}\right))\left(\begin{matrix}0\\16\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-\left(-3\right)}&-\frac{-3}{1-\left(-3\right)}\\-\frac{1}{1-\left(-3\right)}&\frac{1}{1-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}0\\16\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&\frac{3}{4}\\-\frac{1}{4}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}0\\16\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}\times 16\\\frac{1}{4}\times 16\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}12\\4\end{matrix}\right)$$

y va x matritsa elementlarini chiqarib olish.

$$y=12,x=4$$

Chiqarib tashlashdan foydalanish qadamlari

Birinchi tenglamani yeching. Ikkala tarafdan 3x ni ayirish.

$$y-3x=0$$

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.

$$y-3x=0,y+x=16$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali y-3x=0 dan y+x=16 ni ayirish.

$$y-y-3x-x=-16$$

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

$$-3x-x=-16$$

-3x ni -x ga qo'shish.

$$-4x=-16$$

Ikki tarafini -4 ga bo‘ling.

$$x=4$$

4 ni x uchun y+x=16 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz y ni bevosita yecha olasiz.

$$y+4=16$$

Tenglamaning ikkala tarafidan 4 ni ayirish.

$$y=12$$

Tizim hal qilindi.

$$y=12,x=4$$