4x-y-10=0;3x+5y-19=0 eching

$$4 x - y - 10 = 0 ; 3 x + 5 y - 19 = 0$$

x, y uchun yechish

$$x=3$$

$$y=2$$

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.

$$4x-y-10=0,3x+5y-19=0$$

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

$$4x-y-10=0$$

10 ni tenglamaning ikkala tarafiga qo'shish.

$$4x-y=10$$

y ni tenglamaning ikkala tarafiga qo'shish.

$$4x=y+10$$

Ikki tarafini 4 ga bo‘ling.

$$x=\frac{1}{4}\left(y+10\right)$$

\frac{1}{4} ni y+10 marotabaga ko'paytirish.

$$x=\frac{1}{4}y+\frac{5}{2}$$

\frac{y}{4}+\frac{5}{2} ni x uchun boshqa tenglamada almashtirish, 3x+5y-19=0.

$$3\left(\frac{1}{4}y+\frac{5}{2}\right)+5y-19=0$$

3 ni \frac{y}{4}+\frac{5}{2} marotabaga ko'paytirish.

$$\frac{3}{4}y+\frac{15}{2}+5y-19=0$$

\frac{3y}{4} ni 5y ga qo'shish.

$$\frac{23}{4}y+\frac{15}{2}-19=0$$

\frac{15}{2} ni -19 ga qo'shish.

$$\frac{23}{4}y-\frac{23}{2}=0$$

\frac{23}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$\frac{23}{4}y=\frac{23}{2}$$

Tenglamaning ikki tarafini \frac{23}{4} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$y=2$$

2 ni y uchun x=\frac{1}{4}y+\frac{5}{2} da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$x=\frac{1}{4}\times 2+\frac{5}{2}$$

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

$$x=\frac{1+5}{2}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{5}{2} ni \frac{1}{2} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=3$$

Tizim hal qilindi.

$$x=3,y=2$$

Matritsalardan foydalanish qadamlari

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

$$4x-y-10=0,3x+5y-19=0$$

Tenglamalarni matritsa shaklida yozish.

$$\left(\begin{matrix}4&-1\\3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\19\end{matrix}\right)$$

\left(\begin{matrix}4&-1\\3&5\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.

$$inverse(\left(\begin{matrix}4&-1\\3&5\end{matrix}\right))\left(\begin{matrix}4&-1\\3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\3&5\end{matrix}\right))\left(\begin{matrix}10\\19\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\3&5\end{matrix}\right))\left(\begin{matrix}10\\19\end{matrix}\right)$$

Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\3&5\end{matrix}\right))\left(\begin{matrix}10\\19\end{matrix}\right)$$

\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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{4\times 5-\left(-3\right)}&-\frac{-1}{4\times 5-\left(-3\right)}\\-\frac{3}{4\times 5-\left(-3\right)}&\frac{4}{4\times 5-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}10\\19\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{23}&\frac{1}{23}\\-\frac{3}{23}&\frac{4}{23}\end{matrix}\right)\left(\begin{matrix}10\\19\end{matrix}\right)$$

Matritsalarni ko'paytirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{23}\times 10+\frac{1}{23}\times 19\\-\frac{3}{23}\times 10+\frac{4}{23}\times 19\end{matrix}\right)$$

Arifmetik hisobni amalga oshirish.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\2\end{matrix}\right)$$

x va y matritsa elementlarini chiqarib olish.

$$x=3,y=2$$

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.

$$4x-y-10=0,3x+5y-19=0$$

4x va 3x ni teng qilish uchun birinchi tenglamaning har bir tarafida barcha shartlarni 3 ga va ikkinchining har bir tarafidagi barcha shartlarni 4 ga ko'paytiring.

$$3\times 4x+3\left(-1\right)y+3\left(-10\right)=0,4\times 3x+4\times 5y+4\left(-19\right)=0$$

Qisqartirish.

$$12x-3y-30=0,12x+20y-76=0$$

Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali 12x-3y-30=0 dan 12x+20y-76=0 ni ayirish.

$$12x-12x-3y-20y-30+76=0$$

12x ni -12x ga qo'shish. 12x va -12x shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.

$$-3y-20y-30+76=0$$

-3y ni -20y ga qo'shish.

$$-23y-30+76=0$$

-30 ni 76 ga qo'shish.

$$-23y+46=0$$

Tenglamaning ikkala tarafidan 46 ni ayirish.

$$-23y=-46$$

Ikki tarafini -23 ga bo‘ling.

$$y=2$$

2 ni y uchun 3x+5y-19=0 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz x ni bevosita yecha olasiz.

$$3x+5\times 2-19=0$$

5 ni 2 marotabaga ko'paytirish.

$$3x+10-19=0$$

10 ni -19 ga qo'shish.

$$3x-9=0$$

9 ni tenglamaning ikkala tarafiga qo'shish.

$$3x=9$$

Ikki tarafini 3 ga bo‘ling.

$$x=3$$

Tizim hal qilindi.

$$x=3,y=2$$