2x+3y=6quadtext{and}quad6x-5y=4 eching

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

x, y uchun yechish

$$x = \frac{3}{2} = 1\frac{1}{2} = 1,5$$

$$y=1$$

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.

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

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

$$2x+3y=6$$

Tenglamaning ikkala tarafidan 3y ni ayirish.

$$2x=-3y+6$$

Ikki tarafini 2 ga bo‘ling.

$$x=\frac{1}{2}\left(-3y+6\right)$$

\frac{1}{2} ni -3y+6 marotabaga ko'paytirish.

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

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

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

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

$$-9y+18-5y=4$$

-9y ni -5y ga qo'shish.

$$-14y+18=4$$

Tenglamaning ikkala tarafidan 18 ni ayirish.

$$-14y=-14$$

Ikki tarafini -14 ga bo‘ling.

$$y=1$$

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

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

3 ni -\frac{3}{2} ga qo'shish.

$$x=\frac{3}{2}$$

Tizim hal qilindi.

$$x=\frac{3}{2},y=1$$

Matritsalardan foydalanish qadamlari

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

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

Tenglamalarni matritsa shaklida yozish.

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

\left(\begin{matrix}2&3\\6&-5\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

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

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

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

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

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

\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.

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

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{28}&\frac{3}{28}\\\frac{3}{14}&-\frac{1}{14}\end{matrix}\right)\left(\begin{matrix}6\\4\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{28}\times 6+\frac{3}{28}\times 4\\\frac{3}{14}\times 6-\frac{1}{14}\times 4\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

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

x va y matritsa elementlarini chiqarib olish.

$$x=\frac{3}{2},y=1$$

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.

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

2x va 6x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 6 ga va ikkinchining har bir tarafidagi barcha shartlarni 2 ga ko'paytiring.

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

Qisqartirish.

$$12x+18y=36,12x-10y=8$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 12x+18y=36 dan 12x-10y=8 ni ayirish.

$$12x-12x+18y+10y=36-8$$

12x ni -12x ga qo'shish. 12x va -12x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$18y+10y=36-8$$

18y ni 10y ga qo'shish.

$$28y=36-8$$

36 ni -8 ga qo'shish.

$$28y=28$$

Ikki tarafini 28 ga bo‘ling.

$$y=1$$

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

$$6x-5=4$$

5 ni tenglamaning ikkala tarafiga qo'shish.

$$6x=9$$

Ikki tarafini 6 ga bo‘ling.

$$x=\frac{3}{2}$$

Tizim hal qilindi.

$$x=\frac{3}{2},y=1$$