6x+9y=9text{and}-6x+6y=1 eching

$$6 x + 9 y = 9 \text { and } - 6 x + 6 y = 1$$

x, y uchun yechish

$$x=\frac{1}{2}=0,5$$

$$y=\frac{2}{3}\approx 0,666666667$$

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.

$$6x+9y=9,-6x+6y=1$$

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

$$6x+9y=9$$

Tenglamaning ikkala tarafidan 9y ni ayirish.

$$6x=-9y+9$$

Ikki tarafini 6 ga bo‘ling.

$$x=\frac{1}{6}\left(-9y+9\right)$$

\frac{1}{6} ni -9y+9 marotabaga ko'paytirish.

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

\frac{-3y+3}{2} ni x uchun boshqa tenglamada almashtirish, -6x+6y=1.

$$-6\left(-\frac{3}{2}y+\frac{3}{2}\right)+6y=1$$

-6 ni \frac{-3y+3}{2} marotabaga ko'paytirish.

$$9y-9+6y=1$$

9y ni 6y ga qo'shish.

$$15y-9=1$$

9 ni tenglamaning ikkala tarafiga qo'shish.

$$15y=10$$

Ikki tarafini 15 ga bo‘ling.

$$y=\frac{2}{3}$$

\frac{2}{3} ni y uchun x=-\frac{3}{2}y+\frac{3}{2} da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=-\frac{3}{2}\times \frac{2}{3}+\frac{3}{2}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali -\frac{3}{2} ni \frac{2}{3} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

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

\frac{3}{2} ni -1 ga qo'shish.

$$x=\frac{1}{2}$$

Tizim hal qilindi.

$$x=\frac{1}{2},y=\frac{2}{3}$$

Matritsalardan foydalanish qadamlari

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

$$6x+9y=9,-6x+6y=1$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}6&9\\-6&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\1\end{matrix}\right)$$

\left(\begin{matrix}6&9\\-6&6\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}6&9\\-6&6\end{matrix}\right))\left(\begin{matrix}6&9\\-6&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&9\\-6&6\end{matrix}\right))\left(\begin{matrix}9\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&9\\-6&6\end{matrix}\right))\left(\begin{matrix}9\\1\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&9\\-6&6\end{matrix}\right))\left(\begin{matrix}9\\1\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{6\times 6-9\left(-6\right)}&-\frac{9}{6\times 6-9\left(-6\right)}\\-\frac{-6}{6\times 6-9\left(-6\right)}&\frac{6}{6\times 6-9\left(-6\right)}\end{matrix}\right)\left(\begin{matrix}9\\1\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{15}&-\frac{1}{10}\\\frac{1}{15}&\frac{1}{15}\end{matrix}\right)\left(\begin{matrix}9\\1\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{15}\times 9-\frac{1}{10}\\\frac{1}{15}\times 9+\frac{1}{15}\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\\\frac{2}{3}\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=\frac{1}{2},y=\frac{2}{3}$$

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.

$$6x+9y=9,-6x+6y=1$$

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

$$-6\times 6x-6\times 9y=-6\times 9,6\left(-6\right)x+6\times 6y=6$$

Qisqartirish.

$$-36x-54y=-54,-36x+36y=6$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali -36x-54y=-54 dan -36x+36y=6 ni ayirish.

$$-36x+36x-54y-36y=-54-6$$

-36x ni 36x ga qo'shish. -36x va 36x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$-54y-36y=-54-6$$

-54y ni -36y ga qo'shish.

$$-90y=-54-6$$

-54 ni -6 ga qo'shish.

$$-90y=-60$$

Ikki tarafini -90 ga bo‘ling.

$$y=\frac{2}{3}$$

\frac{2}{3} ni y uchun -6x+6y=1 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$-6x+6\times \frac{2}{3}=1$$

6 ni \frac{2}{3} marotabaga ko'paytirish.

$$-6x+4=1$$

Tenglamaning ikkala tarafidan 4 ni ayirish.

$$-6x=-3$$

Ikki tarafini -6 ga bo‘ling.

$$x=\frac{1}{2}$$

Tizim hal qilindi.

$$x=\frac{1}{2},y=\frac{2}{3}$$