Nā nīnau hoʻohālike e kūkākūkā ana i nā ʻōnaehana o nā kaulike linear a me nā kaulike ʻole
He kumuhana koʻikoʻi nā ʻōnaehana o nā kaulike linear a me nā like ʻole i ka makemakika me ka hoʻohana ākea ʻana ma nā ʻano like ʻole, e like me ka hoʻokele waiwai, ka ʻepekema, a me ka ʻenekinia. Ma kēia ʻatikala, e kūkākūkā mākou i nā pilikia hoʻohālike e pili ana i nā ʻōnaehana o nā kaulike linear a me nā like ʻole a pehea e hoʻoponopono ai iā lākou me ka kikoʻī.
Ka Wehewehena o kahi ʻōnaehana o nā kaulike laina
ʻO kahi ʻōnaehana o nā kaulike linear he ʻelua a ʻoi aku paha nā kaulike linear e pili ana kekahi i kekahi. Eia nā hiʻohiʻona:
\[
nā hihia
2x + 3y = 5
4x – y = 1
nā hihia
\]
ʻO ka pahuhopu o ka hoʻoponopono ʻana i kēia ʻōnaehana, ʻo ia ka loaʻa ʻana o nā waiwai o \(x\) a me \(y\) e hoʻokō ana i nā kaulike ʻelua i ka manawa like.
Nā Hana no ka Hoʻoponopono ʻana i nā ʻŌnaehana o nā Kaulike Linear
Aia kekahi mau ʻano hana no ka hoʻoponopono ʻana i nā ʻōnaehana o nā kaulike linear, me:
1. Ke ʻAno Hoʻololi
2. Ke ʻAno Hoʻopau
3. Ke ʻAno Matrix (Inverse a i ʻole Gauss-Jordan)
Laʻana Nīnau 1: Ke ʻAno Hoʻololi
E hoʻoponopono kākou i ka ʻōnaehana ma lalo nei me ka hoʻohana ʻana i ke ʻano pani:
\[
nā hihia
x + 2y = 10
3x – y = 5
nā hihia
\]
ʻanuʻu:
1. E hoʻokaʻawale i kekahi o nā loli i loko o kekahi o nā kaulike.
Mai ka hoʻohālikelike mua, hoʻokaʻawale mākou iā \(x\):
\[
x = 10 – 2y
\]
2. E hoʻololi i ka hōʻike i loaʻa i loko o ka hoohalike ʻē aʻe.
E hoʻololi iā \(x = 10 – 2y\) i loko o ka lua o ka hoohalike:
\[
3(10 – 2y) – y = 5
\]
E hoʻoponopono no \(y\):
\[
30 – 6y – y = 5
\]
\[
30 – 7y = 5
\]
\[
-7y = -25
\]
\[
y = \frac{25}{7}
\]
3. E hoʻohana i nā waiwai i loaʻa e ʻike i nā loli ʻē aʻe.
E hoʻololi iā \(y = \frac{25}{7}\) i loko o ka hōʻike no \(x\):
\[
x = 10 – 2\left(\frac{25}{7}\right)
\]
\[
x = 10 – \frac{50}{7}
\]
\[
x = \frac{70}{7} – \frac{50}{7}
\]
\[
x = \frac{20}{7}
\]
No laila, ʻo nā hopena i ka ʻōnaehana ʻo \( x = \frac{20}{7} \) a me \( y = \frac{25}{7} \).
Laʻana Nīnau 2: ʻAno Hoʻopau
A laila, e hoʻohana kākou i ke ʻano hoʻopau e hoʻoponopono i ka ʻōnaehana penei:
\[
nā hihia
2x + 3y = 12
4x + 6y = 24
nā hihia
\]
Ma kēia hihia, ʻike mākou he nui ka hoohalike ʻelua o ka hoohalike mua. No ka hoʻemi ʻana i ka ʻōnaehana, hiki iā mākou ke hoʻonui i ka hoohalike mua me 2 a laila unuhi iā ia mai ka hoohalike ʻelua:
1. E hoʻonui i ka hoohalike mua me 2:
\[
2(2x + 3y) = 2 \cdot 12
\]
\[
4x + 6y = 24
\]
2. E unuhi i ka hoohalike mua i hoʻonui ʻia mai ka hoohalike ʻelua:
\[
(4x + 6y) – (4x + 6y) = 24 – 24
\]
\[
0 = 0
\]
Hāʻawi kēia iā \(0 = 0\), e hōʻike ana he palena ʻole nā hopena o ka ʻōnaehana a ua hilinaʻi kēia mau kaulike.
Laʻana Nīnau 3: Nā ʻAno like ʻole o ka laina
Ua like nā loina o nā kaulike ʻole linear me nā kaulike linear, akā pili pū me nā hōʻailona kaulike ʻole e like me \(<, \leq, >, \geq\). E nānā kākou i kahi laʻana maʻalahi:
\[
nā hihia
3x – y < 7 \\ 2x + y \geq 4 \end{cases} \] Nā ʻAnuʻu Hana: 1. Hoʻohana mākou i ke ʻano hana kiʻi e hoʻoholo ai i ka ʻāpana hoʻonā o kēia ʻōnaehana. E kaha kiʻi i kēlā me kēia kaulike ʻole. 2. E hoʻololi i ka like ʻole i kahi hoohalike e hoʻoholo ai i ka laina palena: No \(3x - y < 7\), ʻo ka laina palena ʻo \(3x - y = 7\)
