f/11=g/33text{siendo}f+g=40 eching

$$\frac { f } { 11 } = \frac { g } { 33 } \text { siendo } f + g = 40$$

f, g uchun yechish

$$f=10$$

$$g=30$$

O'rnini bosishdan foydalanish qoidalari

Birinchi tenglamani yeching. Tenglamaning ikkala tarafini 33 ga, 11,33 ning eng kichik karralisiga ko‘paytiring.

$$3f=g$$

Ikki tarafini 3 ga bo‘ling.

$$f=\frac{1}{3}g$$

\frac{g}{3} ni f uchun boshqa tenglamada almashtirish, f+g=40.

$$\frac{1}{3}g+g=40$$

\frac{g}{3} ni g ga qo'shish.

$$\frac{4}{3}g=40$$

Tenglamaning ikki tarafini \frac{4}{3} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$g=30$$

30 ni g uchun f=\frac{1}{3}g da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz f ni bevosita yecha olasiz.

$$f=\frac{1}{3}\times 30$$

\frac{1}{3} ni 30 marotabaga ko'paytirish.

$$f=10$$

Tizim hal qilindi.

$$f=10,g=30$$

Matritsalardan foydalanish qadamlari

Birinchi tenglamani yeching. Tenglamaning ikkala tarafini 33 ga, 11,33 ning eng kichik karralisiga ko‘paytiring.

$$3f=g$$

Ikkala tarafdan g ni ayirish.

$$3f-g=0$$

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

$$3f-g=0,f+g=40$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}3&-1\\1&1\end{matrix}\right)\left(\begin{matrix}f\\g\end{matrix}\right)=\left(\begin{matrix}0\\40\end{matrix}\right)$$

\left(\begin{matrix}3&-1\\1&1\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}3&-1\\1&1\end{matrix}\right))\left(\begin{matrix}3&-1\\1&1\end{matrix}\right)\left(\begin{matrix}f\\g\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\1&1\end{matrix}\right))\left(\begin{matrix}0\\40\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}f\\g\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\1&1\end{matrix}\right))\left(\begin{matrix}0\\40\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}f\\g\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\1&1\end{matrix}\right))\left(\begin{matrix}0\\40\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}f\\g\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3-\left(-1\right)}&-\frac{-1}{3-\left(-1\right)}\\-\frac{1}{3-\left(-1\right)}&\frac{3}{3-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}0\\40\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}f\\g\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&\frac{1}{4}\\-\frac{1}{4}&\frac{3}{4}\end{matrix}\right)\left(\begin{matrix}0\\40\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}f\\g\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 40\\\frac{3}{4}\times 40\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}f\\g\end{matrix}\right)=\left(\begin{matrix}10\\30\end{matrix}\right)$$

f va g matritsa elementlarini chiqarib olish.

$$f=10,g=30$$

Chiqarib tashlashdan foydalanish qadamlari

Birinchi tenglamani yeching. Tenglamaning ikkala tarafini 33 ga, 11,33 ning eng kichik karralisiga ko‘paytiring.

$$3f=g$$

Ikkala tarafdan g ni ayirish.

$$3f-g=0$$

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.

$$3f-g=0,f+g=40$$

3f va f ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 1 ga va ikkinchining har bir tarafidagi barcha shartlarni 3 ga ko'paytiring.

$$3f-g=0,3f+3g=3\times 40$$

Qisqartirish.

$$3f-g=0,3f+3g=120$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 3f-g=0 dan 3f+3g=120 ni ayirish.

$$3f-3f-g-3g=-120$$

3f ni -3f ga qo'shish. 3f va -3f shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$-g-3g=-120$$

-g ni -3g ga qo'shish.

$$-4g=-120$$

Ikki tarafini -4 ga bo‘ling.

$$g=30$$

30 ni g uchun f+g=40 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz f ni bevosita yecha olasiz.

$$f+30=40$$

Tenglamaning ikkala tarafidan 30 ni ayirish.

$$f=10$$

Tizim hal qilindi.

$$f=10,g=30$$