4x-5y=-14text{and}7x+y=-5 eching

$$4 x - 5 y = - 14 \text { and } 7 x + y = - 5$$

x, y uchun yechish

$$x=-1$$

$$y=2$$

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.

$$4x-5y=-14,7x+y=-5$$

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

$$4x-5y=-14$$

5y ni tenglamaning ikkala tarafiga qo'shish.

$$4x=5y-14$$

Ikki tarafini 4 ga bo‘ling.

$$x=\frac{1}{4}\left(5y-14\right)$$

\frac{1}{4} ni 5y-14 marotabaga ko'paytirish.

$$x=\frac{5}{4}y-\frac{7}{2}$$

\frac{5y}{4}-\frac{7}{2} ni x uchun boshqa tenglamada almashtirish, 7x+y=-5.

$$7\left(\frac{5}{4}y-\frac{7}{2}\right)+y=-5$$

7 ni \frac{5y}{4}-\frac{7}{2} marotabaga ko'paytirish.

$$\frac{35}{4}y-\frac{49}{2}+y=-5$$

\frac{35y}{4} ni y ga qo'shish.

$$\frac{39}{4}y-\frac{49}{2}=-5$$

\frac{49}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$\frac{39}{4}y=\frac{39}{2}$$

Tenglamaning ikki tarafini \frac{39}{4} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$y=2$$

2 ni y uchun x=\frac{5}{4}y-\frac{7}{2} da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=\frac{5}{4}\times 2-\frac{7}{2}$$

\frac{5}{4} ni 2 marotabaga ko'paytirish.

$$x=\frac{5-7}{2}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{7}{2} ni \frac{5}{2} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=-1$$

Tizim hal qilindi.

$$x=-1,y=2$$

Matritsalardan foydalanish qadamlari

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

$$4x-5y=-14,7x+y=-5$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}4&-5\\7&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-14\\-5\end{matrix}\right)$$

\left(\begin{matrix}4&-5\\7&1\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}4&-5\\7&1\end{matrix}\right))\left(\begin{matrix}4&-5\\7&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\7&1\end{matrix}\right))\left(\begin{matrix}-14\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\7&1\end{matrix}\right))\left(\begin{matrix}-14\\-5\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\7&1\end{matrix}\right))\left(\begin{matrix}-14\\-5\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4-\left(-5\times 7\right)}&-\frac{-5}{4-\left(-5\times 7\right)}\\-\frac{7}{4-\left(-5\times 7\right)}&\frac{4}{4-\left(-5\times 7\right)}\end{matrix}\right)\left(\begin{matrix}-14\\-5\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{39}&\frac{5}{39}\\-\frac{7}{39}&\frac{4}{39}\end{matrix}\right)\left(\begin{matrix}-14\\-5\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{39}\left(-14\right)+\frac{5}{39}\left(-5\right)\\-\frac{7}{39}\left(-14\right)+\frac{4}{39}\left(-5\right)\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

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

x va y matritsa elementlarini chiqarib olish.

$$x=-1,y=2$$

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.

$$4x-5y=-14,7x+y=-5$$

4x va 7x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 7 ga va ikkinchining har bir tarafidagi barcha shartlarni 4 ga ko'paytiring.

$$7\times 4x+7\left(-5\right)y=7\left(-14\right),4\times 7x+4y=4\left(-5\right)$$

Qisqartirish.

$$28x-35y=-98,28x+4y=-20$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 28x-35y=-98 dan 28x+4y=-20 ni ayirish.

$$28x-28x-35y-4y=-98+20$$

28x ni -28x ga qo'shish. 28x va -28x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$-35y-4y=-98+20$$

-35y ni -4y ga qo'shish.

$$-39y=-98+20$$

-98 ni 20 ga qo'shish.

$$-39y=-78$$

Ikki tarafini -39 ga bo‘ling.

$$y=2$$

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

$$7x+2=-5$$

Tenglamaning ikkala tarafidan 2 ni ayirish.

$$7x=-7$$

Ikki tarafini 7 ga bo‘ling.

$$x=-1$$

Tizim hal qilindi.

$$x=-1,y=2$$