︠45b878a4-b595-4a96-bcb8-32cf42110268i︠
%html
Išspręsime lygčių sistemą
\[
\left\{\begin{array}{ll}
x_{1}+x_{2}+2x_{3}+x_{4} &=7\\
3x_{1}+4x_{2}+8x_{3}+5x_{4} &=29\\
x_{1}+3x_{2}+7x_{3}+8x_{4} &=30\\
2x_{1}+2x_{2}+5x_{3}+6x_{4} &=23
\end{array}\right.
\]
Gauso metodu.
︡785af4cd-3e5d-4541-bb2a-cf4bd2dcc03c︡︡{"done":true,"html":"Išspręsime lygčių sistemą\n\\[ \n\\left\\{\\begin{array}{ll}\n x_{1}+x_{2}+2x_{3}+x_{4} &=7\\\\\n 3x_{1}+4x_{2}+8x_{3}+5x_{4} &=29\\\\\n x_{1}+3x_{2}+7x_{3}+8x_{4} &=30\\\\\n 2x_{1}+2x_{2}+5x_{3}+6x_{4} &=23\n\\end{array}\\right.\n\\] \nGauso metodu."}
︠6dcc8c22-2e65-4625-9add-f14a165f442ai︠
%html
Sukuriame sistemos matricą $A$:
︡5234967f-8e2f-4a04-9582-276471284aba︡︡{"done":true,"html":"Sukuriame sistemos matricą $A$:"}
︠75fe8915-986d-4ccd-9e0e-013e759eeeb8s︠
A = matrix(QQ, 4, 4, [[1, 1, 2, 1], [3, 4, 8, 5], [1, 3, 7, 8], [2, 2, 5, 6]])
show(A)
︡10681c8e-e32a-4cf0-b64e-df2429b578c4︡{"html":"
$\\displaystyle \\left(\\begin{array}{rrrr}\n1 & 1 & 2 & 1 \\\\\n3 & 4 & 8 & 5 \\\\\n1 & 3 & 7 & 8 \\\\\n2 & 2 & 5 & 6\n\\end{array}\\right)$
"}︡{"done":true}
︠7f692882-0df4-484c-b5db-4be636e91115︠
︡82cf86f1-6ccc-45ec-9f77-9a9fac5698eb︡
︠72450bd5-2a5c-41ef-9525-44ea8608b70f︠
latex(A)
︡2d6f0c67-7659-42dd-922c-9578be2edffe︡︡{"stdout":"\\left(\\begin{array}{rrrr}\n1 & 1 & 2 & 1 \\\\\n3 & 4 & 8 & 5 \\\\\n1 & 3 & 7 & 8 \\\\\n2 & 2 & 5 & 6\n\\end{array}\\right)\n","done":false}︡{"done":true}
︠fe53683e-5c74-447d-9cda-2e306a6b9c77i︠
%html
Dabar prie matricos prijungiame laisvųjų narių stulpelį:
︡986015fd-dc7e-4d4e-a676-284c6921a828︡︡{"done":true,"html":"Dabar prie matricos prijungiame laisvųjų narių stulpelį:"}
︠69d95535-aba2-474f-8243-74c0b58b21d4s︠
laisvieji_nariai = vector(QQ, [7, 29, 30, 23])
show(laisvieji_nariai)
︡b4285db1-04b9-4a58-b998-56e0367ded21︡{"html":"$\\displaystyle \\left(7,\\,29,\\,30,\\,23\\right)$
"}︡{"done":true}
︠bf5e8a00-6eb1-4525-bdc0-e1ea88f13f30s︠
B = A.augment(laisvieji_nariai, subdivide = True)
show(B)
︡422af654-f8fe-4dbb-85cc-806403e5de94︡{"html":"$\\displaystyle \\left(\\begin{array}{rrrr|r}\n1 & 1 & 2 & 1 & 7 \\\\\n3 & 4 & 8 & 5 & 29 \\\\\n1 & 3 & 7 & 8 & 30 \\\\\n2 & 2 & 5 & 6 & 23\n\\end{array}\\right)$
"}︡{"done":true}
︠440ec02a-4a4f-41ab-a246-a7e64cdda497i︠
%html
Dabar matricai $B$ taikysime elementariąsias eilučių operacijas:
︡8c6175f4-9c7d-4f2c-a2a6-7bc8f92f81a7︡︡{"done":true,"html":"Dabar matricai $B$ taikysime elementariąsias eilučių operacijas:"}
︠01feb7c3-29be-4da4-a800-45baf4d60e55︠
︡d81db65b-c421-4aa9-9f31-491fa621f415︡
︠1a0bb0f7-8a77-4c2c-8ceb-1fb4cd8d8508︠
︡e70f7693-344a-46a0-b4bf-fc235cf10d6c︡
︠4cb91ce4-f90e-452e-abe2-bd987c873c34s︠
B.add_multiple_of_row(1,0,-3)
show(B)
︡d60d2390-2176-4033-9aee-3da6e133d135︡{"html":"$\\displaystyle \\left(\\begin{array}{rrrr|r}\n1 & 1 & 2 & 1 & 7 \\\\\n0 & 1 & 2 & 2 & 8 \\\\\n1 & 3 & 7 & 8 & 30 \\\\\n2 & 2 & 5 & 6 & 23\n\\end{array}\\right)$
"}︡{"done":true}
︠ac06e734-b9de-456a-9b12-6ea58dee25f0s︠
B.add_multiple_of_row(2,0,-1)
show(B)
︡4917c130-48d6-461a-8d34-1c98977be191︡{"html":"$\\displaystyle \\left(\\begin{array}{rrrr|r}\n1 & 1 & 2 & 1 & 7 \\\\\n0 & 1 & 2 & 2 & 8 \\\\\n0 & 2 & 5 & 7 & 23 \\\\\n2 & 2 & 5 & 6 & 23\n\\end{array}\\right)$
"}︡{"done":true}
︠f0e3ef86-ceaa-452b-8032-d78077c1f03bs︠
B.add_multiple_of_row(3,0,-2)
show(B)
︡506795fb-66e2-4b07-9cf8-211ec012d858︡{"html":"$\\displaystyle \\left(\\begin{array}{rrrr|r}\n1 & 1 & 2 & 1 & 7 \\\\\n0 & 1 & 2 & 2 & 8 \\\\\n0 & 2 & 5 & 7 & 23 \\\\\n0 & 0 & 1 & 4 & 9\n\\end{array}\\right)$
"}︡{"done":true}
︠06db7927-4cd1-4995-b877-7a8e61f56eefs︠
B.add_multiple_of_row(2,1,-2)
show(B)
︡9dbdac42-9379-4982-bb4a-dd9c1a7a866e︡{"html":"$\\displaystyle \\left(\\begin{array}{rrrr|r}\n1 & 1 & 2 & 1 & 7 \\\\\n0 & 1 & 2 & 2 & 8 \\\\\n0 & 0 & 1 & 3 & 7 \\\\\n0 & 0 & 1 & 4 & 9\n\\end{array}\\right)$
"}︡{"done":true}
︠8a21fda1-64dc-4bb0-90df-5e287af177d8s︠
B.add_multiple_of_row(3,2,-1)
show(B)
︡1f83a2a8-07e4-4acc-b9bf-ab9f290e62f0︡{"html":"$\\displaystyle \\left(\\begin{array}{rrrr|r}\n1 & 1 & 2 & 1 & 7 \\\\\n0 & 1 & 2 & 2 & 8 \\\\\n0 & 0 & 1 & 3 & 7 \\\\\n0 & 0 & 0 & 1 & 2\n\\end{array}\\right)$
"}︡{"done":true}
︠cadfcd9b-2849-4339-bb7d-3a9ac1a55994s︠
B.add_multiple_of_row(2,3,-3)
show(B)
︡232ad4ce-fa55-4641-b778-bd8fb8ba2960︡{"html":"$\\displaystyle \\left(\\begin{array}{rrrr|r}\n1 & 1 & 2 & 1 & 7 \\\\\n0 & 1 & 2 & 2 & 8 \\\\\n0 & 0 & 1 & 0 & 1 \\\\\n0 & 0 & 0 & 1 & 2\n\\end{array}\\right)$
"}︡{"done":true}︡
︠18131c3c-72ed-497d-9767-94fcef503673s︠
B.add_multiple_of_row(1,3,-2)
show(B)
︡0bcbd396-df19-457b-a155-542e3f5df053︡{"html":"$\\displaystyle \\left(\\begin{array}{rrrr|r}\n1 & 1 & 2 & 1 & 7 \\\\\n0 & 1 & 2 & 0 & 4 \\\\\n0 & 0 & 1 & 3 & 7 \\\\\n0 & 0 & 0 & 1 & 2\n\\end{array}\\right)$
"}︡{"done":true}
︠d06fcd3a-acda-41eb-97e8-0a77fe7591eds︠
B.add_multiple_of_row(0,3,-1)
show(B)
︡b3e146ba-28db-421a-b437-6e2d25744e58︡{"html":"$\\displaystyle \\left(\\begin{array}{rrrr|r}\n1 & 1 & 2 & 0 & 5 \\\\\n0 & 1 & 2 & 0 & 4 \\\\\n0 & 0 & 1 & 0 & 1 \\\\\n0 & 0 & 0 & 1 & 2\n\\end{array}\\right)$
"}︡{"done":true}︡
︠98fb67eb-8b68-467f-a80c-c0db1ae50100s︠
B.add_multiple_of_row(1,2,-2)
show(B)
︡7fa67138-1e2f-4811-b547-0f7bd1b53687︡{"html":"$\\displaystyle \\left(\\begin{array}{rrrr|r}\n1 & 1 & 2 & 0 & 5 \\\\\n0 & 1 & 0 & 0 & 2 \\\\\n0 & 0 & 1 & 0 & 1 \\\\\n0 & 0 & 0 & 1 & 2\n\\end{array}\\right)$
"}︡{"done":true}︡
︠2b1c6ec5-0d17-45a7-91d8-d8a3f77f3207s︠
B.add_multiple_of_row(0,2,-2)
show(B)
︡23f2a648-4c9b-498a-a2b4-06d372ed9ec0︡{"html":"$\\displaystyle \\left(\\begin{array}{rrrr|r}\n1 & 1 & 0 & 0 & 3 \\\\\n0 & 1 & 0 & 0 & 2 \\\\\n0 & 0 & 1 & 0 & 1 \\\\\n0 & 0 & 0 & 1 & 2\n\\end{array}\\right)$
"}︡{"done":true}︡
︠f6a0eac6-ccbe-45bd-acde-d94d7489727fs︠
B.add_multiple_of_row(0,1,-1)
show(B)
︡10b2c688-5806-4fa0-a3e0-a92f50d98625︡{"html":"$\\displaystyle \\left(\\begin{array}{rrrr|r}\n1 & 0 & 0 & 0 & 1 \\\\\n0 & 1 & 0 & 0 & 2 \\\\\n0 & 0 & 1 & 0 & 1 \\\\\n0 & 0 & 0 & 1 & 2\n\\end{array}\\right)$
"}︡{"done":true}︡
︠e4bb53a1-036b-47c1-a8b5-a1973a9a57bai︠
%html
Taigi sistemos sprendinys yra
\[
x_1 = 1,\; x_2 = 2,\; x_3 = 1,\; x_4 = 2.
\]
︡06da9ff2-6aa4-4c5e-a4ca-0aa3354f9382︡︡{"done":true,"html":"Taigi sistemos sprendinys yra \n\\[ \nx_1 = 1,\\; x_2 = 2,\\; x_3 = 1,\\; x_4 = 2.\n\\]"}
︠3488a998-6a97-4f7c-aa4a-abffc30cea8ei︠
%html
Šią lygčių sistemą galima buvo spręsti kitu būdu. Lygčių sistema ekvivalenti tokiai matricų lygčiai:
\[
\left(\begin{array}{rrrr}
1 & 1 & 2 & 1 \\
3 & 4 & 8 & 5 \\
1 & 3 & 7 & 8 \\
2 & 2 & 5 & 6
\end{array}\right)
\left(\begin{array}{r}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{array}\right) =
\left(\begin{array}{r}
7 \\
29 \\
30 \\
23
\end{array}\right)
\]
︡c85a2e9d-6051-478f-96e4-c92ad69a9226︡︡{"done":true,"html":"Šią lygčių sistemą galima buvo spręsti kitu būdu. Lygčių sistema ekvivalenti tokiai matricų lygčiai:\n\\[ \n\\left(\\begin{array}{rrrr}\n1 & 1 & 2 & 1 \\\\\n3 & 4 & 8 & 5 \\\\\n1 & 3 & 7 & 8 \\\\\n2 & 2 & 5 & 6\n\\end{array}\\right) \n\\left(\\begin{array}{r}\nx_1 \\\\\nx_2 \\\\\nx_3 \\\\\nx_4\n\\end{array}\\right) = \n\\left(\\begin{array}{r}\n7 \\\\\n29 \\\\\n30 \\\\\n23\n\\end{array}\\right)\n\\]"}
︠8ac123b3-cf6e-4426-95c3-2489c510c485︠
kintamieji = vector(QQ, [7, 29, 30, 23])
show(laisvieji_nariai)
︠92143d02-7c7d-4e3e-9861-52531d939d8e︠