x/3+y/4=4;x/2-y/4=1 eching

$$\frac { x } { 3 } + \frac { y } { 4 } = 4 ; \frac { x } { 2 } - \frac { y } { 4 } = 1$$

x, y uchun yechish

$$x=6$$

$$y=8$$

O'rnini bosishdan foydalanish qoidalari

Birinchi tenglamani yeching. Tenglamaning ikkala tarafini 12 ga, 3,4 ning eng kichik karralisiga ko‘paytiring.

$$4x+3y=48$$

Ikkinchi tenglamani yeching. Tenglamaning ikkala tarafini 4 ga, 2,4 ning eng kichik karralisiga ko‘paytiring.

$$2x-y=4$$

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.

$$4x+3y=48,2x-y=4$$

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

$$4x+3y=48$$

Tenglamaning ikkala tarafidan 3y ni ayirish.

$$4x=-3y+48$$

Ikki tarafini 4 ga bo‘ling.

$$x=\frac{1}{4}\left(-3y+48\right)$$

\frac{1}{4} ni -3y+48 marotabaga ko'paytirish.

$$x=-\frac{3}{4}y+12$$

-\frac{3y}{4}+12 ni x uchun boshqa tenglamada almashtirish, 2x-y=4.

$$2\left(-\frac{3}{4}y+12\right)-y=4$$

2 ni -\frac{3y}{4}+12 marotabaga ko'paytirish.

$$-\frac{3}{2}y+24-y=4$$

-\frac{3y}{2} ni -y ga qo'shish.

$$-\frac{5}{2}y+24=4$$

Tenglamaning ikkala tarafidan 24 ni ayirish.

$$-\frac{5}{2}y=-20$$

Tenglamaning ikki tarafini -\frac{5}{2} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$y=8$$

8 ni y uchun x=-\frac{3}{4}y+12 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=-\frac{3}{4}\times 8+12$$

-\frac{3}{4} ni 8 marotabaga ko'paytirish.

$$x=-6+12$$

12 ni -6 ga qo'shish.

$$x=6$$

Tizim hal qilindi.

$$x=6,y=8$$

Matritsalardan foydalanish qadamlari

Birinchi tenglamani yeching. Tenglamaning ikkala tarafini 12 ga, 3,4 ning eng kichik karralisiga ko‘paytiring.

$$4x+3y=48$$

Ikkinchi tenglamani yeching. Tenglamaning ikkala tarafini 4 ga, 2,4 ning eng kichik karralisiga ko‘paytiring.

$$2x-y=4$$

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

$$4x+3y=48,2x-y=4$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}4&3\\2&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}48\\4\end{matrix}\right)$$

\left(\begin{matrix}4&3\\2&-1\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}4&3\\2&-1\end{matrix}\right))\left(\begin{matrix}4&3\\2&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\2&-1\end{matrix}\right))\left(\begin{matrix}48\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\2&-1\end{matrix}\right))\left(\begin{matrix}48\\4\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\2&-1\end{matrix}\right))\left(\begin{matrix}48\\4\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4\left(-1\right)-3\times 2}&-\frac{3}{4\left(-1\right)-3\times 2}\\-\frac{2}{4\left(-1\right)-3\times 2}&\frac{4}{4\left(-1\right)-3\times 2}\end{matrix}\right)\left(\begin{matrix}48\\4\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10}&\frac{3}{10}\\\frac{1}{5}&-\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}48\\4\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10}\times 48+\frac{3}{10}\times 4\\\frac{1}{5}\times 48-\frac{2}{5}\times 4\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\8\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=6,y=8$$

Chiqarib tashlashdan foydalanish qadamlari

Birinchi tenglamani yeching. Tenglamaning ikkala tarafini 12 ga, 3,4 ning eng kichik karralisiga ko‘paytiring.

$$4x+3y=48$$

Ikkinchi tenglamani yeching. Tenglamaning ikkala tarafini 4 ga, 2,4 ning eng kichik karralisiga ko‘paytiring.

$$2x-y=4$$

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.

$$4x+3y=48,2x-y=4$$

4x va 2x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 2 ga va ikkinchining har bir tarafidagi barcha shartlarni 4 ga ko'paytiring.

$$2\times 4x+2\times 3y=2\times 48,4\times 2x+4\left(-1\right)y=4\times 4$$

Qisqartirish.

$$8x+6y=96,8x-4y=16$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 8x+6y=96 dan 8x-4y=16 ni ayirish.

$$8x-8x+6y+4y=96-16$$

8x ni -8x ga qo'shish. 8x va -8x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$6y+4y=96-16$$

6y ni 4y ga qo'shish.

$$10y=96-16$$

96 ni -16 ga qo'shish.

$$10y=80$$

Ikki tarafini 10 ga bo‘ling.

$$y=8$$

8 ni y uchun 2x-y=4 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$2x-8=4$$

8 ni tenglamaning ikkala tarafiga qo'shish.

$$2x=12$$

Ikki tarafini 2 ga bo‘ling.

$$x=6$$

Tizim hal qilindi.

$$x=6,y=8$$