y=4/3x-28/3;quady=2x+8 eching

$$y = \frac { 4 } { 3 } x - \frac { 28 } { 3 } ; \quad y = 2 x + 8$$

y, x uchun yechish

$$x=-26$$

$$y=-44$$

O'rnini bosishdan foydalanish qoidalari

Birinchi tenglamani yeching. Ikkala tarafdan \frac{4}{3}x ni ayirish.

$$y-\frac{4}{3}x=-\frac{28}{3}$$

Ikkinchi tenglamani yeching. Ikkala tarafdan 2x ni ayirish.

$$y-2x=8$$

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.

$$y-\frac{4}{3}x=-\frac{28}{3},y-2x=8$$

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

$$y-\frac{4}{3}x=-\frac{28}{3}$$

\frac{4x}{3} ni tenglamaning ikkala tarafiga qo'shish.

$$y=\frac{4}{3}x-\frac{28}{3}$$

\frac{-28+4x}{3} ni y uchun boshqa tenglamada almashtirish, y-2x=8.

$$\frac{4}{3}x-\frac{28}{3}-2x=8$$

\frac{4x}{3} ni -2x ga qo'shish.

$$-\frac{2}{3}x-\frac{28}{3}=8$$

\frac{28}{3} ni tenglamaning ikkala tarafiga qo'shish.

$$-\frac{2}{3}x=\frac{52}{3}$$

Tenglamaning ikki tarafini -\frac{2}{3} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$x=-26$$

-26 ni x uchun y=\frac{4}{3}x-\frac{28}{3} da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz y ni bevosita yecha olasiz.

$$y=\frac{4}{3}\left(-26\right)-\frac{28}{3}$$

\frac{4}{3} ni -26 marotabaga ko'paytirish.

$$y=\frac{-104-28}{3}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{28}{3} ni -\frac{104}{3} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$y=-44$$

Tizim hal qilindi.

$$y=-44,x=-26$$

Matritsalardan foydalanish qadamlari

Birinchi tenglamani yeching. Ikkala tarafdan \frac{4}{3}x ni ayirish.

$$y-\frac{4}{3}x=-\frac{28}{3}$$

Ikkinchi tenglamani yeching. Ikkala tarafdan 2x ni ayirish.

$$y-2x=8$$

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

$$y-\frac{4}{3}x=-\frac{28}{3},y-2x=8$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}1&-\frac{4}{3}\\1&-2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{28}{3}\\8\end{matrix}\right)$$

\left(\begin{matrix}1&-\frac{4}{3}\\1&-2\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}1&-\frac{4}{3}\\1&-2\end{matrix}\right))\left(\begin{matrix}1&-\frac{4}{3}\\1&-2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{4}{3}\\1&-2\end{matrix}\right))\left(\begin{matrix}-\frac{28}{3}\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{4}{3}\\1&-2\end{matrix}\right))\left(\begin{matrix}-\frac{28}{3}\\8\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{4}{3}\\1&-2\end{matrix}\right))\left(\begin{matrix}-\frac{28}{3}\\8\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-\left(-\frac{4}{3}\right)}&-\frac{-\frac{4}{3}}{-2-\left(-\frac{4}{3}\right)}\\-\frac{1}{-2-\left(-\frac{4}{3}\right)}&\frac{1}{-2-\left(-\frac{4}{3}\right)}\end{matrix}\right)\left(\begin{matrix}-\frac{28}{3}\\8\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}3&-2\\\frac{3}{2}&-\frac{3}{2}\end{matrix}\right)\left(\begin{matrix}-\frac{28}{3}\\8\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}3\left(-\frac{28}{3}\right)-2\times 8\\\frac{3}{2}\left(-\frac{28}{3}\right)-\frac{3}{2}\times 8\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-44\\-26\end{matrix}\right)$$

y va x matritsa elementlarini chiqarib olish.

$$y=-44,x=-26$$

Chiqarib tashlashdan foydalanish qadamlari

Birinchi tenglamani yeching. Ikkala tarafdan \frac{4}{3}x ni ayirish.

$$y-\frac{4}{3}x=-\frac{28}{3}$$

Ikkinchi tenglamani yeching. Ikkala tarafdan 2x ni ayirish.

$$y-2x=8$$

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.

$$y-\frac{4}{3}x=-\frac{28}{3},y-2x=8$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali y-\frac{4}{3}x=-\frac{28}{3} dan y-2x=8 ni ayirish.

$$y-y-\frac{4}{3}x+2x=-\frac{28}{3}-8$$

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

$$-\frac{4}{3}x+2x=-\frac{28}{3}-8$$

-\frac{4x}{3} ni 2x ga qo'shish.

$$\frac{2}{3}x=-\frac{28}{3}-8$$

-\frac{28}{3} ni -8 ga qo'shish.

$$\frac{2}{3}x=-\frac{52}{3}$$

Tenglamaning ikki tarafini \frac{2}{3} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$x=-26$$

-26 ni x uchun y-2x=8 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz y ni bevosita yecha olasiz.

$$y-2\left(-26\right)=8$$

-2 ni -26 marotabaga ko'paytirish.

$$y+52=8$$

Tenglamaning ikkala tarafidan 52 ni ayirish.

$$y=-44$$

Tizim hal qilindi.

$$y=-44,x=-26$$