y, x uchun yechish
x=2
y=5
Grafik
Baham ko'rish
Klipbordga nusxa olish
y-2x=1
Birinchi tenglamani yeching. Ikkala tarafdan 2x ni ayirish.
y-2x=1,y+x=7
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-2x=1
Tenglamalardan birini tanlang va teng belgisining chap tomonidagi y ni izolyatsiyalash orqali y ni hisoblang.
y=2x+1
2x ni tenglamaning ikkala tarafiga qo'shish.
2x+1+x=7
2x+1 ni y uchun boshqa tenglamada almashtirish, y+x=7.
3x+1=7
2x ni x ga qo'shish.
3x=6
Tenglamaning ikkala tarafidan 1 ni ayirish.
x=2
Ikki tarafini 3 ga bo‘ling.
y=2\times 2+1
2 ni x uchun y=2x+1 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz y ni bevosita yecha olasiz.
y=4+1
2 ni 2 marotabaga ko'paytirish.
y=5
1 ni 4 ga qo'shish.
y=5,x=2
Tizim hal qilindi.
y-2x=1
Birinchi tenglamani yeching. Ikkala tarafdan 2x ni ayirish.
y-2x=1,y+x=7
Tenglamalar standart shaklda ko'rsatilsin so'ng tenglamalar tizimini yechish uchun matritsalardan foydalanilsin.
\left(\begin{matrix}1&-2\\1&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}1\\7\end{matrix}\right)
Tenglamalarni matritsa shaklida yozish.
inverse(\left(\begin{matrix}1&-2\\1&1\end{matrix}\right))\left(\begin{matrix}1&-2\\1&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\1&1\end{matrix}\right))\left(\begin{matrix}1\\7\end{matrix}\right)
\left(\begin{matrix}1&-2\\1&1\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&-2\\1&1\end{matrix}\right))\left(\begin{matrix}1\\7\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&-2\\1&1\end{matrix}\right))\left(\begin{matrix}1\\7\end{matrix}\right)
Tenglik belgisining chap tomonida matritsalarni koʻpaytiring.
\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-\left(-2\right)}&-\frac{-2}{1-\left(-2\right)}\\-\frac{1}{1-\left(-2\right)}&\frac{1}{1-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}1\\7\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}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&\frac{2}{3}\\-\frac{1}{3}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}1\\7\end{matrix}\right)
Arifmetik hisobni amalga oshirish.
\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}+\frac{2}{3}\times 7\\-\frac{1}{3}+\frac{1}{3}\times 7\end{matrix}\right)
Matritsalarni ko'paytirish.
\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}5\\2\end{matrix}\right)
Arifmetik hisobni amalga oshirish.
y=5,x=2
y va x matritsa elementlarini chiqarib olish.
y-2x=1
Birinchi tenglamani yeching. Ikkala tarafdan 2x ni ayirish.
y-2x=1,y+x=7
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-2x-x=1-7
Har bir teng belgisining yon tarafidan o'sxhash shartlarini ayirish orqali y-2x=1 dan y+x=7 ni ayirish.
-2x-x=1-7
y ni -y ga qo'shish. y va -y shartlari bekor qilinadi va faqatgina yechimi bor bitta o'zgaruvchan qiymat bilan tenglamani tark etadi.
-3x=1-7
-2x ni -x ga qo'shish.
-3x=-6
1 ni -7 ga qo'shish.
x=2
Ikki tarafini -3 ga bo‘ling.
y+2=7
2 ni x uchun y+x=7 da almashtirish. Natija tenglama faqat bitta o'zgaruvchi qiymatga ega bo'lganligi bois siz y ni bevosita yecha olasiz.
y=5
Tenglamaning ikkala tarafidan 2 ni ayirish.
y=5,x=2
Tizim hal qilindi.
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