y=5x+7text{en}y=8x-8 eching

$$y = 5 x + 7 \text { en } y = 8 x - 8$$

y, x uchun yechish

$$x=5$$

$$y=32$$

O'rnini bosishdan foydalanish qoidalari

Birinchi tenglamani yeching. Ikkala tarafdan 5x ni ayirish.

$$y-5x=7$$

Ikkinchi tenglamani yeching. Ikkala tarafdan 8x ni ayirish.

$$y-8x=-8$$

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.

$$y-5x=7,y-8x=-8$$

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

$$y-5x=7$$

5x ni tenglamaning ikkala tarafiga qo'shish.

$$y=5x+7$$

5x+7 ni y uchun boshqa tenglamada almashtirish, y-8x=-8.

$$5x+7-8x=-8$$

5x ni -8x ga qo'shish.

$$-3x+7=-8$$

Tenglamaning ikkala tarafidan 7 ni ayirish.

$$-3x=-15$$

Ikki tarafini -3 ga bo‘ling.

$$x=5$$

5 ni x uchun y=5x+7 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz y ni bevosita yecha olasiz.

$$y=5\times 5+7$$

5 ni 5 marotabaga ko'paytirish.

$$y=25+7$$

7 ni 25 ga qo'shish.

$$y=32$$

Tizim hal qilindi.

$$y=32,x=5$$

Matritsalardan foydalanish qadamlari

Birinchi tenglamani yeching. Ikkala tarafdan 5x ni ayirish.

$$y-5x=7$$

Ikkinchi tenglamani yeching. Ikkala tarafdan 8x ni ayirish.

$$y-8x=-8$$

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

$$y-5x=7,y-8x=-8$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}1&-5\\1&-8\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}7\\-8\end{matrix}\right)$$

\left(\begin{matrix}1&-5\\1&-8\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}1&-5\\1&-8\end{matrix}\right))\left(\begin{matrix}1&-5\\1&-8\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-5\\1&-8\end{matrix}\right))\left(\begin{matrix}7\\-8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-5\\1&-8\end{matrix}\right))\left(\begin{matrix}7\\-8\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-5\\1&-8\end{matrix}\right))\left(\begin{matrix}7\\-8\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{-8-\left(-5\right)}&-\frac{-5}{-8-\left(-5\right)}\\-\frac{1}{-8-\left(-5\right)}&\frac{1}{-8-\left(-5\right)}\end{matrix}\right)\left(\begin{matrix}7\\-8\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{8}{3}&-\frac{5}{3}\\\frac{1}{3}&-\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}7\\-8\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{8}{3}\times 7-\frac{5}{3}\left(-8\right)\\\frac{1}{3}\times 7-\frac{1}{3}\left(-8\right)\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}32\\5\end{matrix}\right)$$

y va x matritsa elementlarini chiqarib olish.

$$y=32,x=5$$

Chiqarib tashlashdan foydalanish qadamlari

Birinchi tenglamani yeching. Ikkala tarafdan 5x ni ayirish.

$$y-5x=7$$

Ikkinchi tenglamani yeching. Ikkala tarafdan 8x ni ayirish.

$$y-8x=-8$$

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.

$$y-5x=7,y-8x=-8$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali y-5x=7 dan y-8x=-8 ni ayirish.

$$y-y-5x+8x=7+8$$

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

$$-5x+8x=7+8$$

-5x ni 8x ga qo'shish.

$$3x=7+8$$

7 ni 8 ga qo'shish.

$$3x=15$$

Ikki tarafini 3 ga bo‘ling.

$$x=5$$

5 ni x uchun y-8x=-8 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz y ni bevosita yecha olasiz.

$$y-8\times 5=-8$$

-8 ni 5 marotabaga ko'paytirish.

$$y-40=-8$$

40 ni tenglamaning ikkala tarafiga qo'shish.

$$y=32$$

Tizim hal qilindi.

$$y=32,x=5$$