x-3y=7text{and}3x+3y=9 eching

$$x - 3 y = 7 \text { and } 3 x + 3 y = 9$$

x, y uchun yechish

$$x=4$$

$$y=-1$$

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.

$$x-3y=7,3x+3y=9$$

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

$$x-3y=7$$

3y ni tenglamaning ikkala tarafiga qo'shish.

$$x=3y+7$$

3y+7 ni x uchun boshqa tenglamada almashtirish, 3x+3y=9.

$$3\left(3y+7\right)+3y=9$$

3 ni 3y+7 marotabaga ko'paytirish.

$$9y+21+3y=9$$

9y ni 3y ga qo'shish.

$$12y+21=9$$

Tenglamaning ikkala tarafidan 21 ni ayirish.

$$12y=-12$$

Ikki tarafini 12 ga bo‘ling.

$$y=-1$$

-1 ni y uchun x=3y+7 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=3\left(-1\right)+7$$

3 ni -1 marotabaga ko'paytirish.

$$x=-3+7$$

7 ni -3 ga qo'shish.

$$x=4$$

Tizim hal qilindi.

$$x=4,y=-1$$

Matritsalardan foydalanish qadamlari

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

$$x-3y=7,3x+3y=9$$

Tenglamalarni matritsa shaklida yozish.

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

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

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

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

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

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

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

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3-\left(-3\times 3\right)}&-\frac{-3}{3-\left(-3\times 3\right)}\\-\frac{3}{3-\left(-3\times 3\right)}&\frac{1}{3-\left(-3\times 3\right)}\end{matrix}\right)\left(\begin{matrix}7\\9\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&\frac{1}{4}\\-\frac{1}{4}&\frac{1}{12}\end{matrix}\right)\left(\begin{matrix}7\\9\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 7+\frac{1}{4}\times 9\\-\frac{1}{4}\times 7+\frac{1}{12}\times 9\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\-1\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=4,y=-1$$

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.

$$x-3y=7,3x+3y=9$$

x va 3x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 3 ga va ikkinchining har bir tarafidagi barcha shartlarni 1 ga ko'paytiring.

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

Qisqartirish.

$$3x-9y=21,3x+3y=9$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 3x-9y=21 dan 3x+3y=9 ni ayirish.

$$3x-3x-9y-3y=21-9$$

3x ni -3x ga qo'shish. 3x va -3x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$-9y-3y=21-9$$

-9y ni -3y ga qo'shish.

$$-12y=21-9$$

21 ni -9 ga qo'shish.

$$-12y=12$$

Ikki tarafini -12 ga bo‘ling.

$$y=-1$$

-1 ni y uchun 3x+3y=9 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$3x+3\left(-1\right)=9$$

3 ni -1 marotabaga ko'paytirish.

$$3x-3=9$$

3 ni tenglamaning ikkala tarafiga qo'shish.

$$3x=12$$

Ikki tarafini 3 ga bo‘ling.

$$x=4$$

Tizim hal qilindi.

$$x=4,y=-1$$