a-m=5text{y}a+m+5=12 eching

$$a - m = 5 \text { y } a + m + 5 = 12$$

a, m uchun yechish

$$a=6$$

$$m=1$$

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.

$$a-m=5,a+m+5=12$$

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

$$a-m=5$$

m ni tenglamaning ikkala tarafiga qo'shish.

$$a=m+5$$

m+5 ni a uchun boshqa tenglamada almashtirish, a+m+5=12.

$$m+5+m+5=12$$

m ni m ga qo'shish.

$$2m+5+5=12$$

5 ni 5 ga qo'shish.

$$2m+10=12$$

Tenglamaning ikkala tarafidan 10 ni ayirish.

$$2m=2$$

Ikki tarafini 2 ga bo‘ling.

$$m=1$$

1 ni m uchun a=m+5 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz a ni bevosita yecha olasiz.

$$a=1+5$$

5 ni 1 ga qo'shish.

$$a=6$$

Tizim hal qilindi.

$$a=6,m=1$$

Matritsalardan foydalanish qadamlari

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

$$a-m=5,a+m+5=12$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}1&-1\\1&1\end{matrix}\right)\left(\begin{matrix}a\\m\end{matrix}\right)=\left(\begin{matrix}5\\7\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-1\\1&1\end{matrix}\right))\left(\begin{matrix}1&-1\\1&1\end{matrix}\right)\left(\begin{matrix}a\\m\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&1\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\m\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&1\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}a\\m\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&1\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}a\\m\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-\left(-1\right)}&-\frac{-1}{1-\left(-1\right)}\\-\frac{1}{1-\left(-1\right)}&\frac{1}{1-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}5\\7\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}a\\m\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{2}\\-\frac{1}{2}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}5\\7\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}a\\m\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 5+\frac{1}{2}\times 7\\-\frac{1}{2}\times 5+\frac{1}{2}\times 7\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}a\\m\end{matrix}\right)=\left(\begin{matrix}6\\1\end{matrix}\right)$$

a va m matritsa elementlarini chiqarib olish.

$$a=6,m=1$$

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.

$$a-m=5,a+m+5=12$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali a-m=5 dan a+m+5=12 ni ayirish.

$$a-a-m-m-5=5-12$$

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

$$-m-m-5=5-12$$

-m ni -m ga qo'shish.

$$-2m-5=5-12$$

5 ni -12 ga qo'shish.

$$-2m-5=-7$$

5 ni tenglamaning ikkala tarafiga qo'shish.

$$-2m=-2$$

Ikki tarafini -2 ga bo‘ling.

$$m=1$$

1 ni m uchun a+m+5=12 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz a ni bevosita yecha olasiz.

$$a+1+5=12$$

1 ni 5 ga qo'shish.

$$a+6=12$$

Tenglamaning ikkala tarafidan 6 ni ayirish.

$$a=6$$

Tizim hal qilindi.

$$a=6,m=1$$