x/5=y/4;y/3=x/3-1 eching

$$\frac { x } { 5 } = \frac { y } { 4 } ; \frac { y } { 3 } = \frac { x } { 3 } - 1$$

x, y uchun yechish

$$x=15$$

$$y=12$$

O'rnini bosishdan foydalanish qoidalari

Birinchi tenglamani yeching. Tenglamaning ikkala tarafini 20 ga, 5,4 ning eng kichik karralisiga ko‘paytiring.

$$4x=5y$$

Ikki tarafini 4 ga bo‘ling.

$$x=\frac{1}{4}\times 5y$$

\frac{1}{4} ni 5y marotabaga ko'paytirish.

$$x=\frac{5}{4}y$$

\frac{5y}{4} ni x uchun boshqa tenglamada almashtirish, -x+y=-3.

$$-\frac{5}{4}y+y=-3$$

-\frac{5y}{4} ni y ga qo'shish.

$$-\frac{1}{4}y=-3$$

Ikkala tarafini -4 ga ko‘paytiring.

$$y=12$$

12 ni y uchun x=\frac{5}{4}y da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=\frac{5}{4}\times 12$$

\frac{5}{4} ni 12 marotabaga ko'paytirish.

$$x=15$$

Tizim hal qilindi.

$$x=15,y=12$$

Matritsalardan foydalanish qadamlari

Birinchi tenglamani yeching. Tenglamaning ikkala tarafini 20 ga, 5,4 ning eng kichik karralisiga ko‘paytiring.

$$4x=5y$$

Ikkala tarafdan 5y ni ayirish.

$$4x-5y=0$$

Ikkinchi tenglamani yeching. Tenglamaning ikkala tarafini 3 ga ko'paytirish.

$$y=x-3$$

Ikkala tarafdan x ni ayirish.

$$y-x=-3$$

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

$$4x-5y=0,-x+y=-3$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}4&-5\\-1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\-3\end{matrix}\right)$$

\left(\begin{matrix}4&-5\\-1&1\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}4&-5\\-1&1\end{matrix}\right))\left(\begin{matrix}4&-5\\-1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\-1&1\end{matrix}\right))\left(\begin{matrix}0\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\-1&1\end{matrix}\right))\left(\begin{matrix}0\\-3\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\-1&1\end{matrix}\right))\left(\begin{matrix}0\\-3\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4-\left(-5\left(-1\right)\right)}&-\frac{-5}{4-\left(-5\left(-1\right)\right)}\\-\frac{-1}{4-\left(-5\left(-1\right)\right)}&\frac{4}{4-\left(-5\left(-1\right)\right)}\end{matrix}\right)\left(\begin{matrix}0\\-3\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&-5\\-1&-4\end{matrix}\right)\left(\begin{matrix}0\\-3\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\left(-3\right)\\-4\left(-3\right)\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\12\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=15,y=12$$

Chiqarib tashlashdan foydalanish qadamlari

Birinchi tenglamani yeching. Tenglamaning ikkala tarafini 20 ga, 5,4 ning eng kichik karralisiga ko‘paytiring.

$$4x=5y$$

Ikkala tarafdan 5y ni ayirish.

$$4x-5y=0$$

Ikkinchi tenglamani yeching. Tenglamaning ikkala tarafini 3 ga ko'paytirish.

$$y=x-3$$

Ikkala tarafdan x ni ayirish.

$$y-x=-3$$

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.

$$4x-5y=0,-x+y=-3$$

4x va -x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni -1 ga va ikkinchining har bir tarafidagi barcha shartlarni 4 ga ko'paytiring.

$$-4x-\left(-5y\right)=0,4\left(-1\right)x+4y=4\left(-3\right)$$

Qisqartirish.

$$-4x+5y=0,-4x+4y=-12$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali -4x+5y=0 dan -4x+4y=-12 ni ayirish.

$$-4x+4x+5y-4y=12$$

-4x ni 4x ga qo'shish. -4x va 4x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$5y-4y=12$$

5y ni -4y ga qo'shish.

$$y=12$$

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

$$-x+12=-3$$

Tenglamaning ikkala tarafidan 12 ni ayirish.

$$-x=-15$$

Ikki tarafini -1 ga bo‘ling.

$$x=15$$

Tizim hal qilindi.

$$x=15,y=12$$