Ngā Pūnaha o ngā Whārite Raina me ngā Taurite Kore: Ngā Ariā, Ngā Whakamahinga, me Ngā Otinga
Ko ngā whārite rārangi me ngā taurite kore e rua ngā ariā matua o te pāngarau e whai wāhi nui ana ki te whānuitanga o ngā marautanga, mai i te ōhanga ki te ahupūngao, mai i te pūtaiao rorohiko ki te koiora. I roto i tēnei tuhinga, ka matapakihia e tātou he aha ngā whārite rārangi me ngā taurite kore e rua, me pēhea te whakaoti i aua whārite, me ō rātou whakamahinga mahi i roto i te oranga o ia rā.
1. Te Whakamāramatanga o ngā Whārite Raina:
Ko te whārite rārangi he whārite e uru ana ki tētahi taurangi he kotahi te mana, he kotahi rānei te nekehanga. Ko te āhua whānui o tētahi whārite rārangi he kotahi te taurangi:
\[ toki + b = 0 \]
ko \(a\) me \(b\) he pūmau, ā, ko \(x\) he taurangi. I tēnei wā, ko te āhua whānui o tētahi whārite rārangi me ngā taurangi e rua:
\[ toki + mā = c \]
ina ko \(a\), \(b\), me \(c\) he pūmau, ā, ko \(x\) me \(y\) he taurangi.
2. Pūnaha Whārite Raina:
Ko te pūnaha whārite rārangi he kohinga o ngā whārite rārangi e rua, neke atu rānei, he ōrite ngā taurangi. Ko ētahi tauira:
\[ \begin{ngā take}
2x + 3y = 6
x – y = 2
\end{ngā take} \]
Ka taea te whakaoti i ēnei pūnaha mā te kimi i ngā uara o ngā taurangi e tutuki ana i ngā whārite katoa o te pūnaha. He maha ngā tikanga hei whakaoti i ngā pūnaha whārite rārangi, tae atu ki te tikanga whakakapinga, te tikanga whakakore, me te tikanga matihiko (te tikanga whakahuri rānei).
3. Tikanga Whakakapinga:
Ko te tikanga whakakapinga ko te whakakapi i tētahi o ngā taurangi ki tētahi kīanga i roto i tētahi atu taurangi. Hei tauira, mō te pūnaha i runga ake nei, ka taea e tātou te whakaoti i te whārite tuarua e pā ana ki \(x\):
\[ x = y + 2 \]
Kātahi ka whakakapia te \( x \) ki te whārite tuatahi:
\[ 2(y + 2) + 3y = 6 \]
I muri i te whakangawari me te whakaoti rapanga, ka taea e tātou te kimi i te uara o \(y\), kātahi ka whakamahi i te uara o \(y\) hei kimi i te \(x\).
4. Tikanga Whakakore:
Ko te tikanga whakakore ko te whakakotahi i ngā whārite hei whakakore i tētahi o ngā taurangi. Ka mahia tēnei mā te tāpiri, te tango rānei i ngā whārite hei whakakore i tētahi taurangi. Hei tauira, ka whakareatia te whārite tuarua ki te 2, kātahi ka tangohia mai i te whārite tuatahi:
\[ 2(x – y) = 4 \Pere Matau 2x – 2y = 4 \]
Tangohia mai i te whārite tuatahi:
\[ (2x + 3y) – (2x – 2y) = 6 – 4 \]
Ka whakamāmā ake tēnei i te:
\[ 5y = 2 \Rightarrow y = \frac{2}{5} \]
Kātahi ka whakakapia te \( y = \frac{2}{5} \) ki roto i tētahi o ngā whārite hei kimi i te \( x \).
5. Tikanga Matrix:
Ko tēnei tikanga he tuhi i te pūnaha whārite ki te āhua matihiko, kātahi ka whakamahi i ngā tikanga taurangi hei kimi i te otinga. Ko te āhua matihiko o te pūnaha whārite i runga ake nei ko:
\[ \begin{pmatrix}
2 me te 3
1 me te -1
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
6
2
\end{pmatrix} \]
Mā te whakamahi i te matihiko whakamuri (mēnā kei te wātea), ka kitea ngā uara o \( x \) me \( y \).
6. Ngā Tauritekore Rārangi:
Ko ngā taurite kore rārangi e pā ana ki tētahi whanaungatanga taurite kore i waenga i ngā kīanga rārangi e rua. Ko te āhua whānui o tētahi taurite kore rārangi me tētahi taurangi ko:
\[ toki + b > 0 \]
\[ toki + b \geq 0 \]
\[ ax + b < 0 \] \[ ax + b \leq 0 \] 7. Ngā Pūnaha o ngā Tauritekore Rārangi: Pērā i ngā whārite rārangi, ko ngā pūnaha o ngā tauritekore rārangi e rua, neke atu rānei ngā tauritekore me te taurangi ōrite. Hei tauira: \[ \begin{cases} 2x + y \leq 5 \\ x - y > 1
\end{ngā take} \]
8. Te Whakatau i ngā Tauritekore Rārangi:
Ko te whakaoti rapanga i tētahi pūnaha o ngā taurite kore rārangi ko te kimi i tētahi huinga otinga e pono ai ngā taurite kore katoa. He maha ngā mahi ka taea e tātou te whai:
– Tuhia ia taunga kore taurite ki ngā taunga Cartesian.
– Tāutuhia te horahanga e tutuki ai ia taurite kore.
– Ko te horahanga e tūtaki ana ki ngā horahanga katoa e tutuki ana, koinei te otinga ki te pūnaha o ngā tauritekore.
9. Te Whakamahinga i te Ao Tūturu:
I roto i te oranga o ia rā, ka puta ake ngā pūnaha whārite rārangi me ngā taurite kore i roto i ngā āhuatanga maha. Anei ētahi tauira:
Ōhanga:
Ko te tātari utu, te arotau hua, me te tātari tono me te tuku he maha ngā wā ka uru atu ki ngā pūnaha whārite rārangi me ngā taurite kore. Hei tauira, i te whakatau i te huinga o ngā hua me whakaputa hei whakanui ake i ngā hua.
Ahupūngao:
Ko ngā ture taketake o te ahupūngao, pērā i ngā ture a Newton, ka tātarihia mā te whakamahi i ngā pūnaha whārite rārangi hei whakatau i te kaha, te papatipu, me te whakaterenga.
Pūtaiao Rorohiko:
Ko ngā rauropi me ō rātou ariā, pērā i te hōtaka rārangi, e whakamahia ana i roto i te hoahoa whatunga, te tohatoha rauemi, me te rangahau whakahaere.
Whakahaere Kaupapa:
Ka taea e te tātari reremahi, te tohatoha rauemi, me te whakahaere wā te whakamahi i ngā taurite kore rārangi hei whakatau i ngā wātaka tino pai.
Koiora:
I roto i te rauropi, he maha ngā wā ka hangaia e ngā tauira taupori he pūnaha whārite rārangi hei mārama ki ngā taunekeneke i waenga i ngā momo me tō rātou taiao.
10. Ngā Wero me ngā Otinga i roto i ngā Pūnaha Raina:
Ahakoa he tino whai hua ngā tikanga kua whakahuatia i runga ake nei, he maha ngā wero kei roto i te whakaoti rapanga i ngā pūnaha whārite rārangi me ngā taurite kore, tae atu ki:
– Te Nui o ngā Whārite me ngā Taurangi: Ina he maha ngā whārite me ngā taurangi o tētahi pūnaha, ka uaua haere ngā tataunga, ā, me whakamahi taputapu rorohiko.
– Te Pūmautanga o te Pūnaha: Kāore he otinga o ngā pūnaha whārite katoa. Ka kore e ōrite te pūnaha mēnā kāore he uara e tutuki ana i ngā whārite katoa.
– Ngā Otinga Maha: He nui atu i te kotahi ngā otinga o ētahi pūnaha (hei tauira, mēnā he hononga rārangi kei waenganui i ngā whārite).
Ko ngā otinga noa ko te whakamahinga o ngā pūmanawa rorohiko me ngā rauropi tau hei whakahaere i ngā pūnaha uaua.
Te Whakamutunga:
He taputapu pāngarau nui ngā pūnaha whārite rārangi me ngā taurite kore mō te tātari i ngā āhuatanga uaua me te whakaoti rapanga o te ao tūturu. Mā te mārama pai ki ō rātou ariā me ngā tikanga whakaoti ka homai he painga ki a tātou i roto i ngā momo mara, ka taea ai e tātou te kimi i ngā otinga tino pai i roto i ngā momo āhuatanga. Me haere tonu te tūhura me te whakaharatau ki te whakaoti i ngā pūnaha rerekē, nā te mea he mea tino nui ēnei pūkenga.
