-3x-3y=3text{and}y=-5x-17 eching

$$- 3 x - 3 y = 3 \text { and } y = - 5 x - 17$$

x, y uchun yechish

$$x=-4$$

$$y=3$$

O'rnini bosishdan foydalanish qoidalari

Ikkinchi tenglamani yeching. 5x ni ikki tarafga qo’shing.

$$y+5x=-17$$

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.

$$-3x-3y=3,5x+y=-17$$

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

$$-3x-3y=3$$

3y ni tenglamaning ikkala tarafiga qo'shish.

$$-3x=3y+3$$

Ikki tarafini -3 ga bo‘ling.

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

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

$$x=-y-1$$

-y-1 ni x uchun boshqa tenglamada almashtirish, 5x+y=-17.

$$5\left(-y-1\right)+y=-17$$

5 ni -y-1 marotabaga ko'paytirish.

$$-5y-5+y=-17$$

-5y ni y ga qo'shish.

$$-4y-5=-17$$

5 ni tenglamaning ikkala tarafiga qo'shish.

$$-4y=-12$$

Ikki tarafini -4 ga bo‘ling.

$$y=3$$

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

$$x=-3-1$$

-1 ni -3 ga qo'shish.

$$x=-4$$

Tizim hal qilindi.

$$x=-4,y=3$$

Matritsalardan foydalanish qadamlari

Ikkinchi tenglamani yeching. 5x ni ikki tarafga qo’shing.

$$y+5x=-17$$

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

$$-3x-3y=3,5x+y=-17$$

Tenglamalarni matritsa shaklida yozish.

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

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

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&-3\\5&1\end{matrix}\right))\left(\begin{matrix}3\\-17\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

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

\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.

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

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{12}&\frac{1}{4}\\-\frac{5}{12}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}3\\-17\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{12}\times 3+\frac{1}{4}\left(-17\right)\\-\frac{5}{12}\times 3-\frac{1}{4}\left(-17\right)\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

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

x va y matritsa elementlarini chiqarib olish.

$$x=-4,y=3$$

Chiqarib tashlashdan foydalanish qadamlari

Ikkinchi tenglamani yeching. 5x ni ikki tarafga qo’shing.

$$y+5x=-17$$

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.

$$-3x-3y=3,5x+y=-17$$

-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.

$$5\left(-3\right)x+5\left(-3\right)y=5\times 3,-3\times 5x-3y=-3\left(-17\right)$$

Qisqartirish.

$$-15x-15y=15,-15x-3y=51$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali -15x-15y=15 dan -15x-3y=51 ni ayirish.

$$-15x+15x-15y+3y=15-51$$

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

$$-15y+3y=15-51$$

-15y ni 3y ga qo'shish.

$$-12y=15-51$$

15 ni -51 ga qo'shish.

$$-12y=-36$$

Ikki tarafini -12 ga bo‘ling.

$$y=3$$

3 ni y uchun 5x+y=-17 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$5x+3=-17$$

Tenglamaning ikkala tarafidan 3 ni ayirish.

$$5x=-20$$

Ikki tarafini 5 ga bo‘ling.

$$x=-4$$

Tizim hal qilindi.

$$x=-4,y=3$$