Te Whakatau i ngā Whārite Wā Kotahi
Ko ngā whārite tukutahi, e mōhiotia ana ko ngā pūnaha whārite, he huinga whārite me ngā taurangi maha. Ka kiia ēnei whārite he tukutahi nā te mea ka whakatauhia ngātahitia, ko te tikanga me tutuki i te otinga ngā whārite katoa i te wā kotahi. Ko te whakaoti whārite tukutahi he pūkenga taketake i roto i te pāngarau e kitea ana ngā tono i roto i ngā momo mara pēnei i te ahupūngao, te ōhanga, te miihini, me te pūtaiao rorohiko. Ko te whāinga o tēnei tuhinga he whakarato i tētahi tirohanga whānui mō ngā tikanga e whakamahia whānuitia ana hei whakaoti whārite tukutahi.
Kaupapa Kaupapa
I mua i te ruku ki ngā tikanga mō te whakaoti rapanga tukutahi, he mea nui kia mārama ki ētahi kupu me ngā ariā matua.
– Ngā Taurangi: Ngā tohu e tohu ana i ngā uara kāore i te mōhiotia, e tohuhia ana e ngā reta pēnei i te x, y, me te z.
– Ngā Whārite Raina: Ngā whārite e whakanuia ana ngā taurangi ki te mana o te kotahi, ā, ka puta he āhua raina (hei tauira, \(2x + 3y = 6\)).
– Ngā Whārite Kore-Rārangi: Ngā whārite e whakauru ana i ngā taurangi kua whakanuia ki ngā mana ehara i te kotahi, e whakauru ana i ngā hua o ngā taurangi, ngā mahi pākoki, me ētahi atu (hei tauira, \(x^2 + y^2 = 9\)).
– Ngā Pūnaha Whārite: He huinga whārite e rua, neke atu rānei, e whai ana i te huinga taurangi ōrite.
Me whakaaro tātou ki tētahi tauira paerewa o tētahi pūnaha whārite rārangi:
\[
\begin{ngā take}
2x + 3y = 6
4x – y = 5
\end{ngā take}
\]
Ko te otinga mō tēnei pūnaha ko tētahi takirua \((x, y)\) e tutuki ana i ngā whārite e rua i te wā kotahi.
Ngā Tikanga mō te Whakaoti i ngā Whārite Wā Kotahi
1. Tikanga Whakairoiro
Ko te tikanga whakairoiro ko te tuhi i ia whārite ki runga i tētahi whatunga taunga me te tautuhi i ngā pūwāhi e tūtaki ai ngā kauwhata. Ko ngā pūwāhi e tūtaki ai ngā whakautu ki te pūnaha whārite.
Hipanga:
1. Tahurihia ia whārite ki te āhua \( y = mx + c \) ko \( m \) te pari, ā, ko \( c \) te taunga-y.
2. Tuhia ngā rārangi e whakaaturia ana e ēnei whārite ki runga i tētahi kauwhata.
3. Kimihia te pūwāhi/ngā pūwāhi whakawhiti o ngā rārangi.
tauira:
Whakaarohia te pūnaha whārite:
\[
\begin{ngā take}
2x + 3y = 6
4x – y = 5
\end{ngā take}
\]
Tahurihia ēnei ki te āhua o te taunga-pikinga:
\[
\begin{ngā take}
y = -\frac{2}{3}x + 2 \\
y = 4x - 5
\end{ngā take}
\]
Tuhia ngā rārangi ki runga i tētahi kauwhata hei kimi i te pūwāhi whakawhiti e tohu ana i te otinga. I tēnei tauira, ko te otinga ko \((x, y) = (1.5, 1)\).
2. Tikanga Whakakapinga
Ko te tikanga whakakapinga ko te whakaoti i tētahi o ngā whārite mō tētahi taurangi me te whakakapi i taua kīanga ki roto i tētahi atu whārite.
Hipanga:
1. Whakaotia tētahi o ngā whārite mō tētahi taurangi.
2. Whakakapia tēnei kīanga ki roto i tētahi atu whārite, ka hua ake he whārite kotahi me te taurangi kotahi.
3. Whakaotia tēnei whārite taurangi-kotahi.
4. Whakahokia te uara i whiwhi ki te kīanga i kitea i te taahiraa 1 hei tiki i te uara o te taurangi tuarua.
tauira:
Whakaarohia te pūnaha:
\[
\begin{ngā take}
2x + 3y = 6
4x – y = 5
\end{ngā take}
\]
Whakaotia te whārite tuatahi mō \( y \):
\[
y = 2 – \frac{2}{3}x
\]
Whakakapia tēnei kīanga ki te whārite tuarua:
\[
4x – (2 – \frac{2}{3}x) = 5
\]
Whakangāwaritia, ka whakaotia hoki mō \( x \):
\[
4x – 2 + \frac{2}{3}x = 5 \\
\frac{14x}{3} = 7 \\
x = \frac{3}{2}
\]
Whakakapia \( x = \frac{3}{2} \) ki roto \( y = 2 – \frac{2}{3}x \):
\[
y = 2 – \frac{2}{3} \times \frac{3}{2} = 2 – 1 = 1
\]
Nō reira, ko te otinga ko \( (x, y) = \left(\frac{3}{2}, 1\right) \).
3. Tikanga Whakakore
Ko te tikanga whakakore ko te tāpiri, te tango rānei i ngā whārite hei whakakore i tētahi taurangi, kia taea ai te whakaoti mō te taurangi e toe ana.
Hipanga:
1. Whakareatia tētahi whārite, e rua rānei ki tētahi pūmau kia rerekē ai ngā taunga o tētahi o ngā taurangi.
2. Tāpirihia, tangohia rānei ngā whārite hei whakakore i tētahi taurangi.
3. Whakaotia te whārite i puta mai mō ngā taurangi e toe ana.
4. Whakakapia tēnei uara ki tētahi o ngā whārite taketake hei kimi i te uara o te taurangi i whakakorea.
tauira:
Whakaarohia te pūnaha:
\[
\begin{ngā take}
2x + 3y = 6
4x – y = 5
\end{ngā take}
\]
Whakareatia te whārite tuarua ki te 3:
\[
4x – y = 5
12x – 3y = 15
\]
Tāpirihia te whārite tuarua kua whakarerekētia ki te whārite tuatahi:
\[
2x + 3y + 12x – 3y = 6 + 15
14x = 21
x = \frac{21}{14} = \frac{3}{2}
\]
Whakakapia \( x = \frac{3}{2} \) ki roto i tētahi o ngā whārite taketake:
\[
2 \left(\frac{3}{2}\right) + 3y = 6 \\
3 + 3y = 6
3y = 3
y = 1
\]
Nō reira, ko te otinga ko \( \left(x, y\right) = \left(\frac{3}{2}, 1\right) \).
4. Tikanga Matrix (Whakakorenga Gaussian me te Whakamuri)
Mō ngā pūnaha nunui ake, ka nui ake te whai hua o ngā tikanga matihiko pēnei i te whakakorenga Gaussian, te whakamahi rānei i te whakamuri matihiko.
Te Whakakorenga Gaussian:
1. Tuhia te pūnaha hei matihiko whakanuia.
2. Whakamahia ngā mahi rarangi hei huri i te matihiko ki te āhua o te taumata rarangi.
3. Whakakapia te whakautu ki muri hei kimi i te otinga.
tauira:
Te punaha:
\[
\begin{ngā take}
2x + 3y = 6
4x – y = 5
\end{ngā take}
\]
Matrix whakanuia:
\[
\begin{pmatrix}
2 me te 3 me te | me te 6 \\
4 me -1 me | me 5
\end{pmatrix}
\]
Ngā mahi rarangi ki te āhua o te rarangi whakarōpū:
1. \( R2 \leftarrow R2 – 2R1 \):
\[
\begin{pmatrix}
2 me te 3 me te | me te 6 \\
0 me -7 me | me -7
\end{pmatrix}
\]
Whakakapinga-muri:
\[
-7y = -7 \e tohu ana ko y = 1 \\
2x + 3(1) = 6 \e kīia ana ko 2x =
