4x-3y=2text{and}x+5y=-11 eching

$$4 x - 3 y = 2 \text { and } x + 5 y = - 11$$

x, y uchun yechish

$$x=-1$$

$$y=-2$$

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.

$$4x-3y=2,x+5y=-11$$

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

$$4x-3y=2$$

3y ni tenglamaning ikkala tarafiga qo'shish.

$$4x=3y+2$$

Ikki tarafini 4 ga bo‘ling.

$$x=\frac{1}{4}\left(3y+2\right)$$

\frac{1}{4} ni 3y+2 marotabaga ko'paytirish.

$$x=\frac{3}{4}y+\frac{1}{2}$$

\frac{3y}{4}+\frac{1}{2} ni x uchun boshqa tenglamada almashtirish, x+5y=-11.

$$\frac{3}{4}y+\frac{1}{2}+5y=-11$$

\frac{3y}{4} ni 5y ga qo'shish.

$$\frac{23}{4}y+\frac{1}{2}=-11$$

Tenglamaning ikkala tarafidan \frac{1}{2} ni ayirish.

$$\frac{23}{4}y=-\frac{23}{2}$$

Tenglamaning ikki tarafini \frac{23}{4} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$y=-2$$

-2 ni y uchun x=\frac{3}{4}y+\frac{1}{2} da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=\frac{3}{4}\left(-2\right)+\frac{1}{2}$$

\frac{3}{4} ni -2 marotabaga ko'paytirish.

$$x=\frac{-3+1}{2}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{2} ni -\frac{3}{2} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=-1$$

Tizim hal qilindi.

$$x=-1,y=-2$$

Matritsalardan foydalanish qadamlari

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

$$4x-3y=2,x+5y=-11$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}4&-3\\1&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\-11\end{matrix}\right)$$

\left(\begin{matrix}4&-3\\1&5\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}4&-3\\1&5\end{matrix}\right))\left(\begin{matrix}4&-3\\1&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-3\\1&5\end{matrix}\right))\left(\begin{matrix}2\\-11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-3\\1&5\end{matrix}\right))\left(\begin{matrix}2\\-11\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-3\\1&5\end{matrix}\right))\left(\begin{matrix}2\\-11\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{4\times 5-\left(-3\right)}&-\frac{-3}{4\times 5-\left(-3\right)}\\-\frac{1}{4\times 5-\left(-3\right)}&\frac{4}{4\times 5-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}2\\-11\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{23}&\frac{3}{23}\\-\frac{1}{23}&\frac{4}{23}\end{matrix}\right)\left(\begin{matrix}2\\-11\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{23}\times 2+\frac{3}{23}\left(-11\right)\\-\frac{1}{23}\times 2+\frac{4}{23}\left(-11\right)\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

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

x va y matritsa elementlarini chiqarib olish.

$$x=-1,y=-2$$

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.

$$4x-3y=2,x+5y=-11$$

4x va x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 1 ga va ikkinchining har bir tarafidagi barcha shartlarni 4 ga ko'paytiring.

$$4x-3y=2,4x+4\times 5y=4\left(-11\right)$$

Qisqartirish.

$$4x-3y=2,4x+20y=-44$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 4x-3y=2 dan 4x+20y=-44 ni ayirish.

$$4x-4x-3y-20y=2+44$$

4x ni -4x ga qo'shish. 4x va -4x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$-3y-20y=2+44$$

-3y ni -20y ga qo'shish.

$$-23y=2+44$$

2 ni 44 ga qo'shish.

$$-23y=46$$

Ikki tarafini -23 ga bo‘ling.

$$y=-2$$

-2 ni y uchun x+5y=-11 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x+5\left(-2\right)=-11$$

5 ni -2 marotabaga ko'paytirish.

$$x-10=-11$$

10 ni tenglamaning ikkala tarafiga qo'shish.

$$x=-1$$

Tizim hal qilindi.

$$x=-1,y=-2$$