3x+2y=8text{and}5x-4y=6 eching

$$3 x + 2 y = 8 \text { and } 5 x - 4 y = 6$$

x, y uchun yechish

$$x=2$$

$$y=1$$

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.

$$3x+2y=8,5x-4y=6$$

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

$$3x+2y=8$$

Tenglamaning ikkala tarafidan 2y ni ayirish.

$$3x=-2y+8$$

Ikki tarafini 3 ga bo‘ling.

$$x=\frac{1}{3}\left(-2y+8\right)$$

\frac{1}{3} ni -2y+8 marotabaga ko'paytirish.

$$x=-\frac{2}{3}y+\frac{8}{3}$$

\frac{-2y+8}{3} ni x uchun boshqa tenglamada almashtirish, 5x-4y=6.

$$5\left(-\frac{2}{3}y+\frac{8}{3}\right)-4y=6$$

5 ni \frac{-2y+8}{3} marotabaga ko'paytirish.

$$-\frac{10}{3}y+\frac{40}{3}-4y=6$$

-\frac{10y}{3} ni -4y ga qo'shish.

$$-\frac{22}{3}y+\frac{40}{3}=6$$

Tenglamaning ikkala tarafidan \frac{40}{3} ni ayirish.

$$-\frac{22}{3}y=-\frac{22}{3}$$

Tenglamaning ikki tarafini -\frac{22}{3} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$y=1$$

1 ni y uchun x=-\frac{2}{3}y+\frac{8}{3} da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=\frac{-2+8}{3}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{8}{3} ni -\frac{2}{3} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=2$$

Tizim hal qilindi.

$$x=2,y=1$$

Matritsalardan foydalanish qadamlari

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

$$3x+2y=8,5x-4y=6$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}3&2\\5&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\6\end{matrix}\right)$$

\left(\begin{matrix}3&2\\5&-4\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}3&2\\5&-4\end{matrix}\right))\left(\begin{matrix}3&2\\5&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\5&-4\end{matrix}\right))\left(\begin{matrix}8\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\5&-4\end{matrix}\right))\left(\begin{matrix}8\\6\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\5&-4\end{matrix}\right))\left(\begin{matrix}8\\6\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{3\left(-4\right)-2\times 5}&-\frac{2}{3\left(-4\right)-2\times 5}\\-\frac{5}{3\left(-4\right)-2\times 5}&\frac{3}{3\left(-4\right)-2\times 5}\end{matrix}\right)\left(\begin{matrix}8\\6\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}&\frac{1}{11}\\\frac{5}{22}&-\frac{3}{22}\end{matrix}\right)\left(\begin{matrix}8\\6\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}\times 8+\frac{1}{11}\times 6\\\frac{5}{22}\times 8-\frac{3}{22}\times 6\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

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

x va y matritsa elementlarini chiqarib olish.

$$x=2,y=1$$

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.

$$3x+2y=8,5x-4y=6$$

3x va 5x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 5 ga va ikkinchining har bir tarafidagi barcha shartlarni 3 ga ko'paytiring.

$$5\times 3x+5\times 2y=5\times 8,3\times 5x+3\left(-4\right)y=3\times 6$$

Qisqartirish.

$$15x+10y=40,15x-12y=18$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 15x+10y=40 dan 15x-12y=18 ni ayirish.

$$15x-15x+10y+12y=40-18$$

15x ni -15x ga qo'shish. 15x va -15x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$10y+12y=40-18$$

10y ni 12y ga qo'shish.

$$22y=40-18$$

40 ni -18 ga qo'shish.

$$22y=22$$

Ikki tarafini 22 ga bo‘ling.

$$y=1$$

1 ni y uchun 5x-4y=6 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$5x-4=6$$

4 ni tenglamaning ikkala tarafiga qo'shish.

$$5x=10$$

Ikki tarafini 5 ga bo‘ling.

$$x=2$$

Tizim hal qilindi.

$$x=2,y=1$$