x:y=15:13text{e}x-y=8 eching

$$x : y = 15 : 13 \text { e } x - y = 8$$

x, y uchun yechish

$$x=60$$

$$y=52$$

O'rnini bosishdan foydalanish qoidalari

Birinchi tenglamani yeching. y qiymati 0 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 13y ga, y,13 ning eng kichik karralisiga ko‘paytiring.

$$13x=15y$$

Ikki tarafini 13 ga bo‘ling.

$$x=\frac{1}{13}\times 15y$$

\frac{1}{13} ni 15y marotabaga ko'paytirish.

$$x=\frac{15}{13}y$$

\frac{15y}{13} ni x uchun boshqa tenglamada almashtirish, x-y=8.

$$\frac{15}{13}y-y=8$$

\frac{15y}{13} ni -y ga qo'shish.

$$\frac{2}{13}y=8$$

Tenglamaning ikki tarafini \frac{2}{13} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$y=52$$

52 ni y uchun x=\frac{15}{13}y da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=\frac{15}{13}\times 52$$

\frac{15}{13} ni 52 marotabaga ko'paytirish.

$$x=60$$

Tizim hal qilindi.

$$x=60,y=52$$

Matritsalardan foydalanish qadamlari

Birinchi tenglamani yeching. y qiymati 0 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 13y ga, y,13 ning eng kichik karralisiga ko‘paytiring.

$$13x=15y$$

Ikkala tarafdan 15y ni ayirish.

$$13x-15y=0$$

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

$$13x-15y=0,x-y=8$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}13&-15\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\8\end{matrix}\right)$$

\left(\begin{matrix}13&-15\\1&-1\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}13&-15\\1&-1\end{matrix}\right))\left(\begin{matrix}13&-15\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}13&-15\\1&-1\end{matrix}\right))\left(\begin{matrix}0\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}13&-15\\1&-1\end{matrix}\right))\left(\begin{matrix}0\\8\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}13&-15\\1&-1\end{matrix}\right))\left(\begin{matrix}0\\8\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{13\left(-1\right)-\left(-15\right)}&-\frac{-15}{13\left(-1\right)-\left(-15\right)}\\-\frac{1}{13\left(-1\right)-\left(-15\right)}&\frac{13}{13\left(-1\right)-\left(-15\right)}\end{matrix}\right)\left(\begin{matrix}0\\8\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}&\frac{15}{2}\\-\frac{1}{2}&\frac{13}{2}\end{matrix}\right)\left(\begin{matrix}0\\8\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{2}\times 8\\\frac{13}{2}\times 8\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}60\\52\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=60,y=52$$

Chiqarib tashlashdan foydalanish qadamlari

Birinchi tenglamani yeching. y qiymati 0 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 13y ga, y,13 ning eng kichik karralisiga ko‘paytiring.

$$13x=15y$$

Ikkala tarafdan 15y ni ayirish.

$$13x-15y=0$$

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.

$$13x-15y=0,x-y=8$$

13x va x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 1 ga va ikkinchining har bir tarafidagi barcha shartlarni 13 ga ko'paytiring.

$$13x-15y=0,13x+13\left(-1\right)y=13\times 8$$

Qisqartirish.

$$13x-15y=0,13x-13y=104$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 13x-15y=0 dan 13x-13y=104 ni ayirish.

$$13x-13x-15y+13y=-104$$

13x ni -13x ga qo'shish. 13x va -13x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$-15y+13y=-104$$

-15y ni 13y ga qo'shish.

$$-2y=-104$$

Ikki tarafini -2 ga bo‘ling.

$$y=52$$

52 ni y uchun x-y=8 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x-52=8$$

52 ni tenglamaning ikkala tarafiga qo'shish.

$$x=60$$

Tizim hal qilindi.

$$x=60,y=52$$