4x-y=5text{and}5y-4x=7 eching

$$4 x - y = 5 \text { and } 5 y - 4 x = 7$$

x, y uchun yechish

$$x=2$$

$$y=3$$

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.

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

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

$$4x-y=5$$

y ni tenglamaning ikkala tarafiga qo'shish.

$$4x=y+5$$

Ikki tarafini 4 ga bo‘ling.

$$x=\frac{1}{4}\left(y+5\right)$$

\frac{1}{4} ni y+5 marotabaga ko'paytirish.

$$x=\frac{1}{4}y+\frac{5}{4}$$

\frac{5+y}{4} ni x uchun boshqa tenglamada almashtirish, -4x+5y=7.

$$-4\left(\frac{1}{4}y+\frac{5}{4}\right)+5y=7$$

-4 ni \frac{5+y}{4} marotabaga ko'paytirish.

$$-y-5+5y=7$$

-y ni 5y ga qo'shish.

$$4y-5=7$$

5 ni tenglamaning ikkala tarafiga qo'shish.

$$4y=12$$

Ikki tarafini 4 ga bo‘ling.

$$y=3$$

3 ni y uchun x=\frac{1}{4}y+\frac{5}{4} da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=\frac{1}{4}\times 3+\frac{5}{4}$$

\frac{1}{4} ni 3 marotabaga ko'paytirish.

$$x=\frac{3+5}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{5}{4} ni \frac{3}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=2$$

Tizim hal qilindi.

$$x=2,y=3$$

Matritsalardan foydalanish qadamlari

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

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

Tenglamalarni matritsa shaklida yozish.

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

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

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

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

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

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

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

\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.

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

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{16}&\frac{1}{16}\\\frac{1}{4}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}5\\7\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{16}\times 5+\frac{1}{16}\times 7\\\frac{1}{4}\times 5+\frac{1}{4}\times 7\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\3\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=2,y=3$$

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.

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

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

$$-4\times 4x-4\left(-1\right)y=-4\times 5,4\left(-4\right)x+4\times 5y=4\times 7$$

Qisqartirish.

$$-16x+4y=-20,-16x+20y=28$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali -16x+4y=-20 dan -16x+20y=28 ni ayirish.

$$-16x+16x+4y-20y=-20-28$$

-16x ni 16x ga qo'shish. -16x va 16x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$4y-20y=-20-28$$

4y ni -20y ga qo'shish.

$$-16y=-20-28$$

-20 ni -28 ga qo'shish.

$$-16y=-48$$

Ikki tarafini -16 ga bo‘ling.

$$y=3$$

3 ni y uchun -4x+5y=7 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$-4x+5\times 3=7$$

5 ni 3 marotabaga ko'paytirish.

$$-4x+15=7$$

Tenglamaning ikkala tarafidan 15 ni ayirish.

$$-4x=-8$$

Ikki tarafini -4 ga bo‘ling.

$$x=2$$

Tizim hal qilindi.

$$x=2,y=3$$