{l}{x=5y+5}{6x-4y=7} eching

$$\left\{ \begin{array} { l } { x = 5y + 5 } \\ { 6 x - 4 y = 7 } \end{array} \right.$$

x, y uchun yechish

$$x=\frac{15}{26}\approx 0,576923077$$

$$y=-\frac{23}{26}\approx -0,884615385$$

O'rnini bosishdan foydalanish qoidalari

Birinchi tenglamani yeching. Ikkala tarafdan 5y ni ayirish.

$$x-5y=5$$

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.

$$x-5y=5,6x-4y=7$$

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

$$x-5y=5$$

5y ni tenglamaning ikkala tarafiga qo'shish.

$$x=5y+5$$

5+5y ni x uchun boshqa tenglamada almashtirish, 6x-4y=7.

$$6\left(5y+5\right)-4y=7$$

6 ni 5+5y marotabaga ko'paytirish.

$$30y+30-4y=7$$

30y ni -4y ga qo'shish.

$$26y+30=7$$

Tenglamaning ikkala tarafidan 30 ni ayirish.

$$26y=-23$$

Ikki tarafini 26 ga bo‘ling.

$$y=-\frac{23}{26}$$

-\frac{23}{26} ni y uchun x=5y+5 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=5\left(-\frac{23}{26}\right)+5$$

5 ni -\frac{23}{26} marotabaga ko'paytirish.

$$x=-\frac{115}{26}+5$$

5 ni -\frac{115}{26} ga qo'shish.

$$x=\frac{15}{26}$$

Tizim hal qilindi.

$$x=\frac{15}{26},y=-\frac{23}{26}$$

Matritsalardan foydalanish qadamlari

Birinchi tenglamani yeching. Ikkala tarafdan 5y ni ayirish.

$$x-5y=5$$

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

$$x-5y=5,6x-4y=7$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}1&-5\\6&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\7\end{matrix}\right)$$

\left(\begin{matrix}1&-5\\6&-4\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}1&-5\\6&-4\end{matrix}\right))\left(\begin{matrix}1&-5\\6&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-5\\6&-4\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-5\\6&-4\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-5\\6&-4\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

2\times 2 matritsasi uchun \left(\begin{matrix}a&b\\c&d\end{matrix}\right), inversiyali matritsa \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), shu bois matritsa tenglamasini matritsaga ko‘paytirish muammosi sifatida qayta yozilishi mumkin.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{-4-\left(-5\times 6\right)}&-\frac{-5}{-4-\left(-5\times 6\right)}\\-\frac{6}{-4-\left(-5\times 6\right)}&\frac{1}{-4-\left(-5\times 6\right)}\end{matrix}\right)\left(\begin{matrix}5\\7\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{13}&\frac{5}{26}\\-\frac{3}{13}&\frac{1}{26}\end{matrix}\right)\left(\begin{matrix}5\\7\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{13}\times 5+\frac{5}{26}\times 7\\-\frac{3}{13}\times 5+\frac{1}{26}\times 7\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{26}\\-\frac{23}{26}\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=\frac{15}{26},y=-\frac{23}{26}$$

Chiqarib tashlashdan foydalanish qadamlari

Birinchi tenglamani yeching. Ikkala tarafdan 5y ni ayirish.

$$x-5y=5$$

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.

$$x-5y=5,6x-4y=7$$

x va 6x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 6 ga va ikkinchining har bir tarafidagi barcha shartlarni 1 ga ko'paytiring.

$$6x+6\left(-5\right)y=6\times 5,6x-4y=7$$

Qisqartirish.

$$6x-30y=30,6x-4y=7$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 6x-30y=30 dan 6x-4y=7 ni ayirish.

$$6x-6x-30y+4y=30-7$$

6x ni -6x ga qo'shish. 6x va -6x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$-30y+4y=30-7$$

-30y ni 4y ga qo'shish.

$$-26y=30-7$$

30 ni -7 ga qo'shish.

$$-26y=23$$

Ikki tarafini -26 ga bo‘ling.

$$y=-\frac{23}{26}$$

-\frac{23}{26} ni y uchun 6x-4y=7 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$6x-4\left(-\frac{23}{26}\right)=7$$

-4 ni -\frac{23}{26} marotabaga ko'paytirish.

$$6x+\frac{46}{13}=7$$

Tenglamaning ikkala tarafidan \frac{46}{13} ni ayirish.

$$6x=\frac{45}{13}$$

Ikki tarafini 6 ga bo‘ling.

$$x=\frac{15}{26}$$

Tizim hal qilindi.

$$x=\frac{15}{26},y=-\frac{23}{26}$$