7x-15y-2=0text{ztal}x+2y=3 eching

$$7 x - 15 y - 2 = 0 \text { ztal } x + 2 y = 3$$

x, y uchun yechish

$$x = \frac{49}{29} = 1\frac{20}{29} \approx 1,689655172$$

$$y=\frac{19}{29}\approx 0,655172414$$

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.

$$7x-15y-2=0,x+2y=3$$

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

$$7x-15y-2=0$$

2 ni tenglamaning ikkala tarafiga qo'shish.

$$7x-15y=2$$

15y ni tenglamaning ikkala tarafiga qo'shish.

$$7x=15y+2$$

Ikki tarafini 7 ga bo‘ling.

$$x=\frac{1}{7}\left(15y+2\right)$$

\frac{1}{7} ni 15y+2 marotabaga ko'paytirish.

$$x=\frac{15}{7}y+\frac{2}{7}$$

\frac{15y+2}{7} ni x uchun boshqa tenglamada almashtirish, x+2y=3.

$$\frac{15}{7}y+\frac{2}{7}+2y=3$$

\frac{15y}{7} ni 2y ga qo'shish.

$$\frac{29}{7}y+\frac{2}{7}=3$$

Tenglamaning ikkala tarafidan \frac{2}{7} ni ayirish.

$$\frac{29}{7}y=\frac{19}{7}$$

Tenglamaning ikki tarafini \frac{29}{7} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$y=\frac{19}{29}$$

\frac{19}{29} ni y uchun x=\frac{15}{7}y+\frac{2}{7} da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=\frac{15}{7}\times \frac{19}{29}+\frac{2}{7}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{15}{7} ni \frac{19}{29} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$x=\frac{285}{203}+\frac{2}{7}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{2}{7} ni \frac{285}{203} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{49}{29}$$

Tizim hal qilindi.

$$x=\frac{49}{29},y=\frac{19}{29}$$

Matritsalardan foydalanish qadamlari

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

$$7x-15y-2=0,x+2y=3$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}7&-15\\1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\3\end{matrix}\right)$$

\left(\begin{matrix}7&-15\\1&2\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}7&-15\\1&2\end{matrix}\right))\left(\begin{matrix}7&-15\\1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-15\\1&2\end{matrix}\right))\left(\begin{matrix}2\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-15\\1&2\end{matrix}\right))\left(\begin{matrix}2\\3\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-15\\1&2\end{matrix}\right))\left(\begin{matrix}2\\3\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7\times 2-\left(-15\right)}&-\frac{-15}{7\times 2-\left(-15\right)}\\-\frac{1}{7\times 2-\left(-15\right)}&\frac{7}{7\times 2-\left(-15\right)}\end{matrix}\right)\left(\begin{matrix}2\\3\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{29}&\frac{15}{29}\\-\frac{1}{29}&\frac{7}{29}\end{matrix}\right)\left(\begin{matrix}2\\3\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{29}\times 2+\frac{15}{29}\times 3\\-\frac{1}{29}\times 2+\frac{7}{29}\times 3\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{49}{29}\\\frac{19}{29}\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=\frac{49}{29},y=\frac{19}{29}$$

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.

$$7x-15y-2=0,x+2y=3$$

7x va x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 1 ga va ikkinchining har bir tarafidagi barcha shartlarni 7 ga ko'paytiring.

$$7x-15y-2=0,7x+7\times 2y=7\times 3$$

Qisqartirish.

$$7x-15y-2=0,7x+14y=21$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 7x-15y-2=0 dan 7x+14y=21 ni ayirish.

$$7x-7x-15y-14y-2=-21$$

7x ni -7x ga qo'shish. 7x va -7x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$-15y-14y-2=-21$$

-15y ni -14y ga qo'shish.

$$-29y-2=-21$$

2 ni tenglamaning ikkala tarafiga qo'shish.

$$-29y=-19$$

Ikki tarafini -29 ga bo‘ling.

$$y=\frac{19}{29}$$

\frac{19}{29} ni y uchun x+2y=3 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x+2\times \frac{19}{29}=3$$

2 ni \frac{19}{29} marotabaga ko'paytirish.

$$x+\frac{38}{29}=3$$

Tenglamaning ikkala tarafidan \frac{38}{29} ni ayirish.

$$x=\frac{49}{29}$$

Tizim hal qilindi.

$$x=\frac{49}{29},y=\frac{19}{29}$$