3x-5y=4;9x-2y=7 eching

$$3 x - 5 y = 4 ; 9 x - 2 y = 7$$

x, y uchun yechish

$$x=\frac{9}{13}\approx 0,692307692$$

$$y=-\frac{5}{13}\approx -0,384615385$$

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.

$$3x-5y=4,9x-2y=7$$

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

$$3x-5y=4$$

5y ni tenglamaning ikkala tarafiga qo'shish.

$$3x=5y+4$$

Ikki tarafini 3 ga bo‘ling.

$$x=\frac{1}{3}\left(5y+4\right)$$

\frac{1}{3} ni 5y+4 marotabaga ko'paytirish.

$$x=\frac{5}{3}y+\frac{4}{3}$$

\frac{5y+4}{3} ni x uchun boshqa tenglamada almashtirish, 9x-2y=7.

$$9\left(\frac{5}{3}y+\frac{4}{3}\right)-2y=7$$

9 ni \frac{5y+4}{3} marotabaga ko'paytirish.

$$15y+12-2y=7$$

15y ni -2y ga qo'shish.

$$13y+12=7$$

Tenglamaning ikkala tarafidan 12 ni ayirish.

$$13y=-5$$

Ikki tarafini 13 ga bo‘ling.

$$y=-\frac{5}{13}$$

-\frac{5}{13} ni y uchun x=\frac{5}{3}y+\frac{4}{3} da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=\frac{5}{3}\left(-\frac{5}{13}\right)+\frac{4}{3}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{5}{3} ni -\frac{5}{13} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$x=-\frac{25}{39}+\frac{4}{3}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{4}{3} ni -\frac{25}{39} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{9}{13}$$

Tizim hal qilindi.

$$x=\frac{9}{13},y=-\frac{5}{13}$$

Matritsalardan foydalanish qadamlari

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

$$3x-5y=4,9x-2y=7$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}3&-5\\9&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\7\end{matrix}\right)$$

\left(\begin{matrix}3&-5\\9&-2\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}3&-5\\9&-2\end{matrix}\right))\left(\begin{matrix}3&-5\\9&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\9&-2\end{matrix}\right))\left(\begin{matrix}4\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\9&-2\end{matrix}\right))\left(\begin{matrix}4\\7\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\9&-2\end{matrix}\right))\left(\begin{matrix}4\\7\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{3\left(-2\right)-\left(-5\times 9\right)}&-\frac{-5}{3\left(-2\right)-\left(-5\times 9\right)}\\-\frac{9}{3\left(-2\right)-\left(-5\times 9\right)}&\frac{3}{3\left(-2\right)-\left(-5\times 9\right)}\end{matrix}\right)\left(\begin{matrix}4\\7\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{39}&\frac{5}{39}\\-\frac{3}{13}&\frac{1}{13}\end{matrix}\right)\left(\begin{matrix}4\\7\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{39}\times 4+\frac{5}{39}\times 7\\-\frac{3}{13}\times 4+\frac{1}{13}\times 7\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{13}\\-\frac{5}{13}\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=\frac{9}{13},y=-\frac{5}{13}$$

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.

$$3x-5y=4,9x-2y=7$$

3x va 9x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 9 ga va ikkinchining har bir tarafidagi barcha shartlarni 3 ga ko'paytiring.

$$9\times 3x+9\left(-5\right)y=9\times 4,3\times 9x+3\left(-2\right)y=3\times 7$$

Qisqartirish.

$$27x-45y=36,27x-6y=21$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 27x-45y=36 dan 27x-6y=21 ni ayirish.

$$27x-27x-45y+6y=36-21$$

27x ni -27x ga qo'shish. 27x va -27x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$-45y+6y=36-21$$

-45y ni 6y ga qo'shish.

$$-39y=36-21$$

36 ni -21 ga qo'shish.

$$-39y=15$$

Ikki tarafini -39 ga bo‘ling.

$$y=-\frac{5}{13}$$

-\frac{5}{13} ni y uchun 9x-2y=7 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$9x-2\left(-\frac{5}{13}\right)=7$$

-2 ni -\frac{5}{13} marotabaga ko'paytirish.

$$9x+\frac{10}{13}=7$$

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

$$9x=\frac{81}{13}$$

Ikki tarafini 9 ga bo‘ling.

$$x=\frac{9}{13}$$

Tizim hal qilindi.

$$x=\frac{9}{13},y=-\frac{5}{13}$$