ax+3y=15text{ve}3x+by=4d eching

$$a x + 3 y = 15 \text { ve } 3 x + b y = 4 d$$

x, y uchun yechish (complex solution)

$$\left\{\begin{matrix}x=-\frac{3\left(5b-4d\right)}{9-ab}\text{, }y=\frac{45-4ad}{9-ab}\text{, }&b=0\text{ or }a\neq \frac{9}{b}\\x\in \mathrm{C}\text{, }y=\frac{15-ax}{3}\text{, }&d=\frac{45}{4a}\text{ and }b=\frac{9}{a}\text{ and }a\neq 0\end{matrix}\right,$$

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.

$$ax+3y=15,3x+by=4d$$

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

$$ax+3y=15$$

Tenglamaning ikkala tarafidan 3y ni ayirish.

$$ax=-3y+15$$

Ikki tarafini a ga bo‘ling.

$$x=\frac{1}{a}\left(-3y+15\right)$$

\frac{1}{a} ni -3y+15 marotabaga ko'paytirish.

$$x=\left(-\frac{3}{a}\right)y+\frac{15}{a}$$

\frac{3\left(5-y\right)}{a} ni x uchun boshqa tenglamada almashtirish, 3x+by=4d.

$$3\left(\left(-\frac{3}{a}\right)y+\frac{15}{a}\right)+by=4d$$

3 ni \frac{3\left(5-y\right)}{a} marotabaga ko'paytirish.

$$\left(-\frac{9}{a}\right)y+\frac{45}{a}+by=4d$$

-\frac{9y}{a} ni by ga qo'shish.

$$\left(b-\frac{9}{a}\right)y+\frac{45}{a}=4d$$

Tenglamaning ikkala tarafidan \frac{45}{a} ni ayirish.

$$\left(b-\frac{9}{a}\right)y=4d-\frac{45}{a}$$

Ikki tarafini b-\frac{9}{a} ga bo‘ling.

$$y=\frac{4ad-45}{ab-9}$$

\frac{4da-45}{ba-9} ni y uchun x=\left(-\frac{3}{a}\right)y+\frac{15}{a} da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=\left(-\frac{3}{a}\right)\times \frac{4ad-45}{ab-9}+\frac{15}{a}$$

-\frac{3}{a} ni \frac{4da-45}{ba-9} marotabaga ko'paytirish.

$$x=-\frac{3\left(4ad-45\right)}{a\left(ab-9\right)}+\frac{15}{a}$$

\frac{15}{a} ni -\frac{3\left(4da-45\right)}{a\left(ba-9\right)} ga qo'shish.

$$x=\frac{3\left(5b-4d\right)}{ab-9}$$

Tizim hal qilindi.

$$x=\frac{3\left(5b-4d\right)}{ab-9},y=\frac{4ad-45}{ab-9}$$

Matritsalardan foydalanish qadamlari

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

$$ax+3y=15,3x+by=4d$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}a&3\\3&b\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\4d\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}a&3\\3&b\end{matrix}\right))\left(\begin{matrix}a&3\\3&b\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}a&3\\3&b\end{matrix}\right))\left(\begin{matrix}15\\4d\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}a&3\\3&b\end{matrix}\right))\left(\begin{matrix}15\\4d\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}a&3\\3&b\end{matrix}\right))\left(\begin{matrix}15\\4d\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{b}{ab-3\times 3}&-\frac{3}{ab-3\times 3}\\-\frac{3}{ab-3\times 3}&\frac{a}{ab-3\times 3}\end{matrix}\right)\left(\begin{matrix}15\\4d\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{b}{ab-9}&-\frac{3}{ab-9}\\-\frac{3}{ab-9}&\frac{a}{ab-9}\end{matrix}\right)\left(\begin{matrix}15\\4d\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{b}{ab-9}\times 15+\left(-\frac{3}{ab-9}\right)\times 4d\\\left(-\frac{3}{ab-9}\right)\times 15+\frac{a}{ab-9}\times 4d\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3\left(5b-4d\right)}{ab-9}\\\frac{4ad-45}{ab-9}\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=\frac{3\left(5b-4d\right)}{ab-9},y=\frac{4ad-45}{ab-9}$$

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.

$$ax+3y=15,3x+by=4d$$

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

$$3ax+3\times 3y=3\times 15,a\times 3x+aby=a\times 4d$$

Qisqartirish.

$$3ax+9y=45,3ax+aby=4ad$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 3ax+9y=45 dan 3ax+aby=4ad ni ayirish.

$$3ax+\left(-3a\right)x+9y+\left(-ab\right)y=45-4ad$$

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

$$9y+\left(-ab\right)y=45-4ad$$

9y ni -aby ga qo'shish.

$$\left(9-ab\right)y=45-4ad$$

Ikki tarafini 9-ab ga bo‘ling.

$$y=\frac{45-4ad}{9-ab}$$

\frac{45-4ad}{9-ab} ni y uchun 3x+by=4d da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$3x+b\times \frac{45-4ad}{9-ab}=4d$$

b ni \frac{45-4ad}{9-ab} marotabaga ko'paytirish.

$$3x+\frac{b\left(45-4ad\right)}{9-ab}=4d$$

Tenglamaning ikkala tarafidan \frac{b\left(45-4ad\right)}{9-ab} ni ayirish.

$$3x=\frac{9\left(4d-5b\right)}{9-ab}$$

Ikki tarafini 3 ga bo‘ling.

$$x=\frac{3\left(4d-5b\right)}{9-ab}$$

Tizim hal qilindi.

$$x=\frac{3\left(4d-5b\right)}{9-ab},y=\frac{45-4ad}{9-ab}$$

x, y uchun yechish

$$\left\{\begin{matrix}x=-\frac{3\left(5b-4d\right)}{9-ab}\text{, }y=\frac{45-4ad}{9-ab}\text{, }&b=0\text{ or }a\neq \frac{9}{b}\\x\in \mathrm{R}\text{, }y=\frac{15-ax}{3}\text{, }&d=\frac{45}{4a}\text{ and }b=\frac{9}{a}\text{ and }a\neq 0\end{matrix}\right,$$

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.

$$ax+3y=15,3x+by=4d$$

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

$$ax+3y=15$$

Tenglamaning ikkala tarafidan 3y ni ayirish.

$$ax=-3y+15$$

Ikki tarafini a ga bo‘ling.

$$x=\frac{1}{a}\left(-3y+15\right)$$

\frac{1}{a} ni -3y+15 marotabaga ko'paytirish.

$$x=\left(-\frac{3}{a}\right)y+\frac{15}{a}$$

\frac{3\left(5-y\right)}{a} ni x uchun boshqa tenglamada almashtirish, 3x+by=4d.

$$3\left(\left(-\frac{3}{a}\right)y+\frac{15}{a}\right)+by=4d$$

3 ni \frac{3\left(5-y\right)}{a} marotabaga ko'paytirish.

$$\left(-\frac{9}{a}\right)y+\frac{45}{a}+by=4d$$

-\frac{9y}{a} ni by ga qo'shish.

$$\left(b-\frac{9}{a}\right)y+\frac{45}{a}=4d$$

Tenglamaning ikkala tarafidan \frac{45}{a} ni ayirish.

$$\left(b-\frac{9}{a}\right)y=4d-\frac{45}{a}$$

Ikki tarafini b-\frac{9}{a} ga bo‘ling.

$$y=\frac{4ad-45}{ab-9}$$

\frac{4da-45}{ba-9} ni y uchun x=\left(-\frac{3}{a}\right)y+\frac{15}{a} da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=\left(-\frac{3}{a}\right)\times \frac{4ad-45}{ab-9}+\frac{15}{a}$$

-\frac{3}{a} ni \frac{4da-45}{ba-9} marotabaga ko'paytirish.

$$x=-\frac{3\left(4ad-45\right)}{a\left(ab-9\right)}+\frac{15}{a}$$

\frac{15}{a} ni -\frac{3\left(4da-45\right)}{a\left(ba-9\right)} ga qo'shish.

$$x=\frac{3\left(5b-4d\right)}{ab-9}$$

Tizim hal qilindi.

$$x=\frac{3\left(5b-4d\right)}{ab-9},y=\frac{4ad-45}{ab-9}$$

Matritsalardan foydalanish qadamlari

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

$$ax+3y=15,3x+by=4d$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}a&3\\3&b\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\4d\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}a&3\\3&b\end{matrix}\right))\left(\begin{matrix}a&3\\3&b\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}a&3\\3&b\end{matrix}\right))\left(\begin{matrix}15\\4d\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}a&3\\3&b\end{matrix}\right))\left(\begin{matrix}15\\4d\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}a&3\\3&b\end{matrix}\right))\left(\begin{matrix}15\\4d\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{b}{ab-3\times 3}&-\frac{3}{ab-3\times 3}\\-\frac{3}{ab-3\times 3}&\frac{a}{ab-3\times 3}\end{matrix}\right)\left(\begin{matrix}15\\4d\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{b}{ab-9}&-\frac{3}{ab-9}\\-\frac{3}{ab-9}&\frac{a}{ab-9}\end{matrix}\right)\left(\begin{matrix}15\\4d\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{b}{ab-9}\times 15+\left(-\frac{3}{ab-9}\right)\times 4d\\\left(-\frac{3}{ab-9}\right)\times 15+\frac{a}{ab-9}\times 4d\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3\left(5b-4d\right)}{ab-9}\\\frac{4ad-45}{ab-9}\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=\frac{3\left(5b-4d\right)}{ab-9},y=\frac{4ad-45}{ab-9}$$

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.

$$ax+3y=15,3x+by=4d$$

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

$$3ax+3\times 3y=3\times 15,a\times 3x+aby=a\times 4d$$

Qisqartirish.

$$3ax+9y=45,3ax+aby=4ad$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 3ax+9y=45 dan 3ax+aby=4ad ni ayirish.

$$3ax+\left(-3a\right)x+9y+\left(-ab\right)y=45-4ad$$

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

$$9y+\left(-ab\right)y=45-4ad$$

9y ni -aby ga qo'shish.

$$\left(9-ab\right)y=45-4ad$$

Ikki tarafini 9-ab ga bo‘ling.

$$y=\frac{45-4ad}{9-ab}$$

\frac{45-4ad}{9-ab} ni y uchun 3x+by=4d da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$3x+b\times \frac{45-4ad}{9-ab}=4d$$

b ni \frac{45-4ad}{9-ab} marotabaga ko'paytirish.

$$3x+\frac{b\left(45-4ad\right)}{9-ab}=4d$$

Tenglamaning ikkala tarafidan \frac{b\left(45-4ad\right)}{9-ab} ni ayirish.

$$3x=\frac{9\left(4d-5b\right)}{9-ab}$$

Ikki tarafini 3 ga bo‘ling.

$$x=\frac{3\left(4d-5b\right)}{9-ab}$$

Tizim hal qilindi.

$$x=\frac{3\left(4d-5b\right)}{9-ab},y=\frac{45-4ad}{9-ab}$$