x+3y=6;5x-2y=13 eching

$$x + 3 y = 6 ; 5 x - 2 y = 13$$

x, y uchun yechish

$$x=3$$

$$y=1$$

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.

$$x+3y=6,5x-2y=13$$

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

$$x+3y=6$$

Tenglamaning ikkala tarafidan 3y ni ayirish.

$$x=-3y+6$$

-3y+6 ni x uchun boshqa tenglamada almashtirish, 5x-2y=13.

$$5\left(-3y+6\right)-2y=13$$

5 ni -3y+6 marotabaga ko'paytirish.

$$-15y+30-2y=13$$

-15y ni -2y ga qo'shish.

$$-17y+30=13$$

Tenglamaning ikkala tarafidan 30 ni ayirish.

$$-17y=-17$$

Ikki tarafini -17 ga bo‘ling.

$$y=1$$

1 ni y uchun x=-3y+6 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=-3+6$$

6 ni -3 ga qo'shish.

$$x=3$$

Tizim hal qilindi.

$$x=3,y=1$$

Matritsalardan foydalanish qadamlari

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

$$x+3y=6,5x-2y=13$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}1&3\\5&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\13\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&3\\5&-2\end{matrix}\right))\left(\begin{matrix}1&3\\5&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\5&-2\end{matrix}\right))\left(\begin{matrix}6\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\5&-2\end{matrix}\right))\left(\begin{matrix}6\\13\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&3\\5&-2\end{matrix}\right))\left(\begin{matrix}6\\13\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-3\times 5}&-\frac{3}{-2-3\times 5}\\-\frac{5}{-2-3\times 5}&\frac{1}{-2-3\times 5}\end{matrix}\right)\left(\begin{matrix}6\\13\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{17}&\frac{3}{17}\\\frac{5}{17}&-\frac{1}{17}\end{matrix}\right)\left(\begin{matrix}6\\13\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{17}\times 6+\frac{3}{17}\times 13\\\frac{5}{17}\times 6-\frac{1}{17}\times 13\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

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

x va y matritsa elementlarini chiqarib olish.

$$x=3,y=1$$

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.

$$x+3y=6,5x-2y=13$$

x va 5x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 5 ga va ikkinchining har bir tarafidagi barcha shartlarni 1 ga ko'paytiring.

$$5x+5\times 3y=5\times 6,5x-2y=13$$

Qisqartirish.

$$5x+15y=30,5x-2y=13$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 5x+15y=30 dan 5x-2y=13 ni ayirish.

$$5x-5x+15y+2y=30-13$$

5x ni -5x ga qo'shish. 5x va -5x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$15y+2y=30-13$$

15y ni 2y ga qo'shish.

$$17y=30-13$$

30 ni -13 ga qo'shish.

$$17y=17$$

Ikki tarafini 17 ga bo‘ling.

$$y=1$$

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

$$5x-2=13$$

2 ni tenglamaning ikkala tarafiga qo'shish.

$$5x=15$$

Ikki tarafini 5 ga bo‘ling.

$$x=3$$

Tizim hal qilindi.

$$x=3,y=1$$