4x+2y=0quadtext{D)}6x-2y=0 eching

$$4 x + 2 y = 0 \quad \text { D) } 6 x - 2 y = 0$$

x, y uchun yechish

$$x=0$$

$$y=0$$

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.

$$4x+2y=0,6x-2y=0$$

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

$$4x+2y=0$$

Tenglamaning ikkala tarafidan 2y ni ayirish.

$$4x=-2y$$

Ikki tarafini 4 ga bo‘ling.

$$x=\frac{1}{4}\left(-2\right)y$$

\frac{1}{4} ni -2y marotabaga ko'paytirish.

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

-\frac{y}{2} ni x uchun boshqa tenglamada almashtirish, 6x-2y=0.

$$6\left(-\frac{1}{2}\right)y-2y=0$$

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

$$-3y-2y=0$$

-3y ni -2y ga qo'shish.

$$-5y=0$$

Ikki tarafini -5 ga bo‘ling.

$$y=0$$

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

$$x=0$$

Tizim hal qilindi.

$$x=0,y=0$$

Matritsalardan foydalanish qadamlari

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

$$4x+2y=0,6x-2y=0$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}4&2\\6&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\0\end{matrix}\right)$$

\left(\begin{matrix}4&2\\6&-2\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}4&2\\6&-2\end{matrix}\right))\left(\begin{matrix}4&2\\6&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\6&-2\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\6&-2\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\6&-2\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{4\left(-2\right)-2\times 6}&-\frac{2}{4\left(-2\right)-2\times 6}\\-\frac{6}{4\left(-2\right)-2\times 6}&\frac{4}{4\left(-2\right)-2\times 6}\end{matrix}\right)\left(\begin{matrix}0\\0\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10}&\frac{1}{10}\\\frac{3}{10}&-\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}0\\0\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\0\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=0,y=0$$

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.

$$4x+2y=0,6x-2y=0$$

4x va 6x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 6 ga va ikkinchining har bir tarafidagi barcha shartlarni 4 ga ko'paytiring.

$$6\times 4x+6\times 2y=0,4\times 6x+4\left(-2\right)y=0$$

Qisqartirish.

$$24x+12y=0,24x-8y=0$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 24x+12y=0 dan 24x-8y=0 ni ayirish.

$$24x-24x+12y+8y=0$$

24x ni -24x ga qo'shish. 24x va -24x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$12y+8y=0$$

12y ni 8y ga qo'shish.

$$20y=0$$

Ikki tarafini 20 ga bo‘ling.

$$y=0$$

0 ni y uchun 6x-2y=0 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$6x=0$$

Ikki tarafini 6 ga bo‘ling.

$$x=0$$

Tizim hal qilindi.

$$x=0,y=0$$