x/7+y/3=5;-x/14+3y=26 eching

$$\frac { x } { 7 } + \frac { y } { 3 } = 5 ; - \frac { x } { 14 } + 3 y = 26$$

x, y uchun yechish

$$x=14$$

$$y=9$$

O'rnini bosishdan foydalanish qoidalari

Birinchi tenglamani yeching. Tenglamaning ikkala tarafini 21 ga, 7,3 ning eng kichik karralisiga ko‘paytiring.

$$3x+7y=105$$

Ikkinchi tenglamani yeching. Tenglamaning ikkala tarafini 14 ga ko'paytirish.

$$-x+42y=364$$

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.

$$3x+7y=105,-x+42y=364$$

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

$$3x+7y=105$$

Tenglamaning ikkala tarafidan 7y ni ayirish.

$$3x=-7y+105$$

Ikki tarafini 3 ga bo‘ling.

$$x=\frac{1}{3}\left(-7y+105\right)$$

\frac{1}{3} ni -7y+105 marotabaga ko'paytirish.

$$x=-\frac{7}{3}y+35$$

-\frac{7y}{3}+35 ni x uchun boshqa tenglamada almashtirish, -x+42y=364.

$$-\left(-\frac{7}{3}y+35\right)+42y=364$$

-1 ni -\frac{7y}{3}+35 marotabaga ko'paytirish.

$$\frac{7}{3}y-35+42y=364$$

\frac{7y}{3} ni 42y ga qo'shish.

$$\frac{133}{3}y-35=364$$

35 ni tenglamaning ikkala tarafiga qo'shish.

$$\frac{133}{3}y=399$$

Tenglamaning ikki tarafini \frac{133}{3} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$y=9$$

9 ni y uchun x=-\frac{7}{3}y+35 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=-\frac{7}{3}\times 9+35$$

-\frac{7}{3} ni 9 marotabaga ko'paytirish.

$$x=-21+35$$

35 ni -21 ga qo'shish.

$$x=14$$

Tizim hal qilindi.

$$x=14,y=9$$

Matritsalardan foydalanish qadamlari

Birinchi tenglamani yeching. Tenglamaning ikkala tarafini 21 ga, 7,3 ning eng kichik karralisiga ko‘paytiring.

$$3x+7y=105$$

Ikkinchi tenglamani yeching. Tenglamaning ikkala tarafini 14 ga ko'paytirish.

$$-x+42y=364$$

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

$$3x+7y=105,-x+42y=364$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}3&7\\-1&42\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}105\\364\end{matrix}\right)$$

\left(\begin{matrix}3&7\\-1&42\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}3&7\\-1&42\end{matrix}\right))\left(\begin{matrix}3&7\\-1&42\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&7\\-1&42\end{matrix}\right))\left(\begin{matrix}105\\364\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&7\\-1&42\end{matrix}\right))\left(\begin{matrix}105\\364\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&7\\-1&42\end{matrix}\right))\left(\begin{matrix}105\\364\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{42}{3\times 42-7\left(-1\right)}&-\frac{7}{3\times 42-7\left(-1\right)}\\-\frac{-1}{3\times 42-7\left(-1\right)}&\frac{3}{3\times 42-7\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}105\\364\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{19}&-\frac{1}{19}\\\frac{1}{133}&\frac{3}{133}\end{matrix}\right)\left(\begin{matrix}105\\364\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{19}\times 105-\frac{1}{19}\times 364\\\frac{1}{133}\times 105+\frac{3}{133}\times 364\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14\\9\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=14,y=9$$

Chiqarib tashlashdan foydalanish qadamlari

Birinchi tenglamani yeching. Tenglamaning ikkala tarafini 21 ga, 7,3 ning eng kichik karralisiga ko‘paytiring.

$$3x+7y=105$$

Ikkinchi tenglamani yeching. Tenglamaning ikkala tarafini 14 ga ko'paytirish.

$$-x+42y=364$$

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.

$$3x+7y=105,-x+42y=364$$

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

$$-3x-7y=-105,3\left(-1\right)x+3\times 42y=3\times 364$$

Qisqartirish.

$$-3x-7y=-105,-3x+126y=1092$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali -3x-7y=-105 dan -3x+126y=1092 ni ayirish.

$$-3x+3x-7y-126y=-105-1092$$

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

$$-7y-126y=-105-1092$$

-7y ni -126y ga qo'shish.

$$-133y=-105-1092$$

-105 ni -1092 ga qo'shish.

$$-133y=-1197$$

Ikki tarafini -133 ga bo‘ling.

$$y=9$$

9 ni y uchun -x+42y=364 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$-x+42\times 9=364$$

42 ni 9 marotabaga ko'paytirish.

$$-x+378=364$$

Tenglamaning ikkala tarafidan 378 ni ayirish.

$$-x=-14$$

Ikki tarafini -1 ga bo‘ling.

$$x=14$$

Tizim hal qilindi.

$$x=14,y=9$$