2x-3y+10=0quadtext{KaHa}5x-y+4=0 eching

$$2 x - 3 y + 10 = 0 \quad \text { KaHa } 5 x - y + 4 = 0$$

x, y uchun yechish

$$x=-\frac{2}{13}\approx -0,153846154$$

$$y = \frac{42}{13} = 3\frac{3}{13} \approx 3,230769231$$

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.

$$2x-3y+10=0,5x-y+4=0$$

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

$$2x-3y+10=0$$

Tenglamaning ikkala tarafidan 10 ni ayirish.

$$2x-3y=-10$$

3y ni tenglamaning ikkala tarafiga qo'shish.

$$2x=3y-10$$

Ikki tarafini 2 ga bo‘ling.

$$x=\frac{1}{2}\left(3y-10\right)$$

\frac{1}{2} ni 3y-10 marotabaga ko'paytirish.

$$x=\frac{3}{2}y-5$$

\frac{3y}{2}-5 ni x uchun boshqa tenglamada almashtirish, 5x-y+4=0.

$$5\left(\frac{3}{2}y-5\right)-y+4=0$$

5 ni \frac{3y}{2}-5 marotabaga ko'paytirish.

$$\frac{15}{2}y-25-y+4=0$$

\frac{15y}{2} ni -y ga qo'shish.

$$\frac{13}{2}y-25+4=0$$

-25 ni 4 ga qo'shish.

$$\frac{13}{2}y-21=0$$

21 ni tenglamaning ikkala tarafiga qo'shish.

$$\frac{13}{2}y=21$$

Tenglamaning ikki tarafini \frac{13}{2} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$y=\frac{42}{13}$$

\frac{42}{13} ni y uchun x=\frac{3}{2}y-5 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=\frac{3}{2}\times \frac{42}{13}-5$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{3}{2} ni \frac{42}{13} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$x=\frac{63}{13}-5$$

-5 ni \frac{63}{13} ga qo'shish.

$$x=-\frac{2}{13}$$

Tizim hal qilindi.

$$x=-\frac{2}{13},y=\frac{42}{13}$$

Matritsalardan foydalanish qadamlari

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

$$2x-3y+10=0,5x-y+4=0$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}2&-3\\5&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-10\\-4\end{matrix}\right)$$

\left(\begin{matrix}2&-3\\5&-1\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}2&-3\\5&-1\end{matrix}\right))\left(\begin{matrix}2&-3\\5&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\5&-1\end{matrix}\right))\left(\begin{matrix}-10\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\5&-1\end{matrix}\right))\left(\begin{matrix}-10\\-4\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\5&-1\end{matrix}\right))\left(\begin{matrix}-10\\-4\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2\left(-1\right)-\left(-3\times 5\right)}&-\frac{-3}{2\left(-1\right)-\left(-3\times 5\right)}\\-\frac{5}{2\left(-1\right)-\left(-3\times 5\right)}&\frac{2}{2\left(-1\right)-\left(-3\times 5\right)}\end{matrix}\right)\left(\begin{matrix}-10\\-4\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{13}&\frac{3}{13}\\-\frac{5}{13}&\frac{2}{13}\end{matrix}\right)\left(\begin{matrix}-10\\-4\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{13}\left(-10\right)+\frac{3}{13}\left(-4\right)\\-\frac{5}{13}\left(-10\right)+\frac{2}{13}\left(-4\right)\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{13}\\\frac{42}{13}\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=-\frac{2}{13},y=\frac{42}{13}$$

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.

$$2x-3y+10=0,5x-y+4=0$$

2x va 5x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 5 ga va ikkinchining har bir tarafidagi barcha shartlarni 2 ga ko'paytiring.

$$5\times 2x+5\left(-3\right)y+5\times 10=0,2\times 5x+2\left(-1\right)y+2\times 4=0$$

Qisqartirish.

$$10x-15y+50=0,10x-2y+8=0$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 10x-15y+50=0 dan 10x-2y+8=0 ni ayirish.

$$10x-10x-15y+2y+50-8=0$$

10x ni -10x ga qo'shish. 10x va -10x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$-15y+2y+50-8=0$$

-15y ni 2y ga qo'shish.

$$-13y+50-8=0$$

50 ni -8 ga qo'shish.

$$-13y+42=0$$

Tenglamaning ikkala tarafidan 42 ni ayirish.

$$-13y=-42$$

Ikki tarafini -13 ga bo‘ling.

$$y=\frac{42}{13}$$

\frac{42}{13} ni y uchun 5x-y+4=0 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$5x-\frac{42}{13}+4=0$$

-\frac{42}{13} ni 4 ga qo'shish.

$$5x+\frac{10}{13}=0$$

Tenglamaning ikkala tarafidan \frac{10}{13} ni ayirish.

$$5x=-\frac{10}{13}$$

Ikki tarafini 5 ga bo‘ling.

$$x=-\frac{2}{13}$$

Tizim hal qilindi.

$$x=-\frac{2}{13},y=\frac{42}{13}$$