2x+y=1text{and}x-y=-4 eching

$$2 x + y = 1 \text { and } x - y = - 4$$

x, y uchun yechish

$$x=-1$$

$$y=3$$

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.

$$2x+y=1,x-y=-4$$

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

$$2x+y=1$$

Tenglamaning ikkala tarafidan y ni ayirish.

$$2x=-y+1$$

Ikki tarafini 2 ga bo‘ling.

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

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

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

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

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

-\frac{y}{2} ni -y ga qo'shish.

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

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

$$-\frac{3}{2}y=-\frac{9}{2}$$

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

$$y=3$$

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

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

-\frac{1}{2} ni 3 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.

$$x=-1,y=3$$

Matritsalardan foydalanish qadamlari

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

$$2x+y=1,x-y=-4$$

Tenglamalarni matritsa shaklida yozish.

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

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

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

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

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

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

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

\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.

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

Arifmetik hisobni amalga oshirish.

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

Matritsalarni ko'paytirish.

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

Arifmetik hisobni amalga oshirish.

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

x va y matritsa elementlarini chiqarib olish.

$$x=-1,y=3$$

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.

$$2x+y=1,x-y=-4$$

2x va x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 1 ga va ikkinchining har bir tarafidagi barcha shartlarni 2 ga ko'paytiring.

$$2x+y=1,2x+2\left(-1\right)y=2\left(-4\right)$$

Qisqartirish.

$$2x+y=1,2x-2y=-8$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 2x+y=1 dan 2x-2y=-8 ni ayirish.

$$2x-2x+y+2y=1+8$$

2x ni -2x ga qo'shish. 2x va -2x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$y+2y=1+8$$

y ni 2y ga qo'shish.

$$3y=1+8$$

1 ni 8 ga qo'shish.

$$3y=9$$

Ikki tarafini 3 ga bo‘ling.

$$y=3$$

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

$$x-3=-4$$

3 ni tenglamaning ikkala tarafiga qo'shish.

$$x=-1$$

Tizim hal qilindi.

$$x=-1,y=3$$