y, x uchun yechish
x=2
y=-3
Grafik
Baham ko'rish
Klipbordga nusxa olish
y+\frac{3}{2}x=0
Birinchi tenglamani yeching. \frac{3}{2}x ni ikki tarafga qo’shing.
y+\frac{1}{2}x=-2
Ikkinchi tenglamani yeching. \frac{1}{2}x ni ikki tarafga qo’shing.
y+\frac{3}{2}x=0,y+\frac{1}{2}x=-2
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.
y+\frac{3}{2}x=0
Tenglamalardan birini tanlang va teng belgisining chap tomonidagi y ni izolyatsiyalash orqali y ni hisoblang.
y=-\frac{3}{2}x
Tenglamaning ikkala tarafidan \frac{3x}{2} ni ayirish.
-\frac{3}{2}x+\frac{1}{2}x=-2
-\frac{3x}{2} ni y uchun boshqa tenglamada almashtirish, y+\frac{1}{2}x=-2.
-x=-2
-\frac{3x}{2} ni \frac{x}{2} ga qo'shish.
x=2
Ikki tarafini -1 ga bo‘ling.
y=-\frac{3}{2}\times 2
2 ni x uchun y=-\frac{3}{2}x da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz y ni bevosita yecha olasiz.
y=-3
-\frac{3}{2} ni 2 marotabaga ko'paytirish.
y=-3,x=2
Tizim hal qilindi.
y+\frac{3}{2}x=0
Birinchi tenglamani yeching. \frac{3}{2}x ni ikki tarafga qo’shing.
y+\frac{1}{2}x=-2
Ikkinchi tenglamani yeching. \frac{1}{2}x ni ikki tarafga qo’shing.
y+\frac{3}{2}x=0,y+\frac{1}{2}x=-2
Tenglamalar standart shaklda ko'rsatilsin so'ng tenglamalar tizimini yechish uchun matritsalardan foydalanilsin.
\left(\begin{matrix}1&\frac{3}{2}\\1&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\-2\end{matrix}\right)
Tenglamalarni matritsa shaklida yozish.
inverse(\left(\begin{matrix}1&\frac{3}{2}\\1&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}1&\frac{3}{2}\\1&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&\frac{3}{2}\\1&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}0\\-2\end{matrix}\right)
\left(\begin{matrix}1&\frac{3}{2}\\1&\frac{1}{2}\end{matrix}\right) teskari matritsasi bilan tenglamani chapdan ko‘paytiring.
\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&\frac{3}{2}\\1&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}0\\-2\end{matrix}\right)
Matritsaning ko‘paytmasi va teskarisi o‘zaro teng matristsadir.
\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&\frac{3}{2}\\1&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}0\\-2\end{matrix}\right)
Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.
\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{1}{2}}{\frac{1}{2}-\frac{3}{2}}&-\frac{\frac{3}{2}}{\frac{1}{2}-\frac{3}{2}}\\-\frac{1}{\frac{1}{2}-\frac{3}{2}}&\frac{1}{\frac{1}{2}-\frac{3}{2}}\end{matrix}\right)\left(\begin{matrix}0\\-2\end{matrix}\right)
2\times 2 matritsasi uchun \left(\begin{matrix}a&b\\c&d\end{matrix}\right), inversiyali matritsa \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), shu bois matritsa tenglamasini matritsaga ko‘paytirish muammosi sifatida qayta yozilishi mumkin.
\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}&\frac{3}{2}\\1&-1\end{matrix}\right)\left(\begin{matrix}0\\-2\end{matrix}\right)
Arifmetik hisobni amalga oshirish.
\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2}\left(-2\right)\\-\left(-2\right)\end{matrix}\right)
Matritsalarni ko'paytirish.
\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-3\\2\end{matrix}\right)
Arifmetik hisobni amalga oshirish.
y=-3,x=2
y va x matritsa elementlarini chiqarib olish.
y+\frac{3}{2}x=0
Birinchi tenglamani yeching. \frac{3}{2}x ni ikki tarafga qo’shing.
y+\frac{1}{2}x=-2
Ikkinchi tenglamani yeching. \frac{1}{2}x ni ikki tarafga qo’shing.
y+\frac{3}{2}x=0,y+\frac{1}{2}x=-2
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.
y-y+\frac{3}{2}x-\frac{1}{2}x=2
Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali y+\frac{3}{2}x=0 dan y+\frac{1}{2}x=-2 ni ayirish.
\frac{3}{2}x-\frac{1}{2}x=2
y ni -y ga qo'shish. y va -y shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.
x=2
\frac{3x}{2} ni -\frac{x}{2} ga qo'shish.
y+\frac{1}{2}\times 2=-2
2 ni x uchun y+\frac{1}{2}x=-2 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz y ni bevosita yecha olasiz.
y+1=-2
\frac{1}{2} ni 2 marotabaga ko'paytirish.
y=-3
Tenglamaning ikkala tarafidan 1 ni ayirish.
y=-3,x=2
Tizim hal qilindi.
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