Te Ariā o ngā Whārite Raina
He ariā taketake ngā whārite rārangi i roto i te pāngarau, ā, he maha ngā whakamahinga i roto i te pūtaiao, te hangarau, te ōhanga, me te maha atu o ngā mara. Ko te mārama ki ngā whārite rārangi te mea nui ki te whakaoti rapanga o te ao tūturu e pā ana ki ngā whanaungatanga rārangi i waenga i ngā taurangi. Ka whakamāramahia e tēnei tuhinga te ariā o ngā whārite rārangi, me pēhea te whakaoti, me ētahi o ā rātou whakamahinga mahi.
Te Whakamāramatanga o ngā Whārite Raina
Ko te whārite rārangi he whārite e whai wāhi ana tētahi, neke atu rānei o ngā taurangi, ā, ko te mana teitei rawa o te taurangi he kotahi. Ko te āhua whānui o tētahi whārite rārangi me tētahi taurangi ka taea te tuhi penei:
\[ toki + b = 0 \]
ina he pūmau a \( a \) me \( b \) , ā, he taurangi a \( x \) .
Mō tētahi whārite rārangi me ngā taurangi e rua, ko te āhua whānui koia tēnei:
\[ toki + mā + c = 0 \]
ina ko \( a \), \( b \), me \( c \) he pūmau, ā, ko \( x \) me \( y \) he taurangi.
I te horopaki whānui, ka taea e ngā whārite rārangi te whakauru i ngā taurangi neke atu i te rua, ā, ka taea hoki te tuhi hei puka matihiko.
Ngā Tauira o ngā Whārite Raina i roto i te Taurangi Kotahi
Whakaarohia te whārite:
\[ 3x – 5 = 0 \]
Hei whakaoti i tēnei, me kimi e tātou te uara o \( x \) e tika ai te whārite. I tēnei wā, ka nukuhia te pūmau ki te taha matau o te whārite:
\[ 3x = 5 \]
Kātahi, wehea ngā taha e rua mā te tauwehenga o \( x \):
\[ x = \frac{5}{3} \]
Nō reira, ko te otinga ki te whārite \( 3x – 5 = 0 \) ko \( x = \frac{5}{3} \).
Ngā Tauira o ngā Whārite Raina i roto i ngā Taurangi e Rua
Whakaarohia te whārite:
\[ 2x + 3y – 6 = 0 \]
E whakaahua ana tēnei whārite i tētahi rārangi i roto i tētahi papa Cartesian rua-ahu. Hei whakaahua i tēnei rārangi, ka kitea ōna pūwāhi e whakawhiti ana ki te tuaka-x me te tuaka-y.
Mō te taunga-x (koinei te \( y = 0 \)):
\[ 2x – 6 = 0 \]
\[ 2x = 6 \]
\[ x = 3 \]
Mō te taunga-y (koinei te \( x = 0 \)):
\[ 3y – 6 = 0 \]
\[ 3y = 6 \]
\[ y = 2 \]
Nō reira, ka haere tēnei rārangi mā roto i ngā pūwāhi (3, 0) me (0, 2).
Te Whakatau i tētahi Pūnaha Whārite Raina
He maha ngā wā ka tūtaki tātou ki ngā pūnaha whārite rārangi, arā, he kohinga whārite rārangi me whakaoti i te wā kotahi. He maha ngā tikanga ka taea te whakamahi hei whakaoti i ngā pūnaha whārite rārangi, tae atu ki:
1. Tikanga Whakakapinga
Ko te tikanga whakakapinga ko te whakaoti i tētahi o ngā whārite mō tētahi taurangi, kātahi ka whakakapi i te hua ki tētahi atu whārite. Hei tauira, whakaarohia te pūnaha whārite e whai ake nei:
\[ 2x + y = 5 \]
\[ x – 2y = -4 \]
Tuatahi, ka whakaotihia e tātou te whārite tuatahi mō \( y \):
\[ y = 5 – 2x \]
Kātahi ka whakakapia te \( y \) ki te whārite tuarua:
\[ x – 2(5 – 2x) = -4 \]
\[ x – 10 + 4x = -4 \]
\[ 5x – 10 = -4 \]
\[ 5x = 6 \]
\[ x = \frac{6}{5} \]
Kātahi ka whakakapia te uara o \( x \) ki roto i te whārite \( y = 5 – 2x \):
\[ y = 5 – 2\left( \frac{6}{5} \right) \]
\[ y = 5 – \frac{12}{5} \]
\[ y = \frac{25}{5} – \frac{12}{5} \]
\[ y = \frac{13}{5} \]
Nō reira, ko te otinga mō te pūnaha whārite ko \( x = \frac{6}{5} \) me \( y = \frac{13}{5} \).
2. Tikanga Whakakore
Ko te tikanga whakakore he tāpiri, he tango rānei i ngā whārite hei whakakore i tētahi o ngā taurangi. Whakaarohia te pūnaha whārite:
\[ 3x + 2y = 8 \]
\[ 2x – 3y = -1 \]
Hei whakakore i te \( y \), ka taea e tātou te tāpiri i ngā whārite i muri i te whakarea i ia whārite ki te tauwehenga e tika ana:
Whakareatia te whārite tuatahi ki te 3, me te whārite tuarua ki te 2:
\[ 9x + 6y = 24 \]
\[ 4x – 6y = -2 \]
Kātahi ka tāpirihia ngā whārite e rua:
\[ 13x = 22 \]
\[ x = \frac{22}{13} \]
Whakakapia te uara o \( x \) ki roto i tētahi o ngā whārite taketake hei kimi i te \( y \):
\[ 3\left( \frac{22}{13} \right) + 2y = 8 \]
\[ \frac{66}{13} + 2y = 8 \]
\[ 2y = 8 – \frac{66}{13} \]
\[ 2y = \frac{104}{13} – \frac{66}{13} \]
\[ 2y = \frac{38}{13} \]
\[ y = \frac{19}{13} \]
Nō reira, ko te otinga mō te pūnaha whārite ko \( x = \frac{22}{13} \) me \( y = \frac{19}{13} \).
3. Tikanga Matrix (Whakakorenga Gaussian)
I tēnei tikanga, ka whakamahia e mātou ngā matihiko hei whakahaere i te pūnaha o ngā whārite kia taea ai te whakaoti i aua whārite i roto i tētahi huarahi pūnaha ake. Hei tauira, hei whakaoti i te pūnaha:
\[ 3x + 2y = 8 \]
\[ 2x – 3y = -1 \]
Ka taea e tātou te tuhi i roto i te puka matihiko whakanui:
\[ \begin{pmatrix}
3 me te 2 me te | me te 8
2 me -3 me | me -1
\end{pmatrix} \]
Ko te mahi e whai ake nei ko te whakamahi i ngā mahi rarangi taketake hei whakaoti i tēnei pūnaha. Heoi, i te mea he uaua ngā taipitopito o tēnei tikanga, me nui ake te ako hōhonu kia mārama ai.
4. Tikanga Whakairoiro
Mā te tikanga whakairoiro ka taea e tātou te kimi otinga mā te tuhi i te whārite ki runga i te papa taunga me te kimi i ngā pūwāhi whakawhiti o ngā kauwhata. Hei tauira, mō te pūnaha:
\[ y = 2x + 1 \]
\[ y = -x + 3 \]
Ka tuhia ēnei rārangi e rua ki te papa xy, ka whakatauhia te pūwāhi e tutaki ai ngā rārangi e rua, koinei te otinga ki te pūnaha whārite.
Ngā Whakamahinga o ngā Whārite Raina
He whānuitia ngā whakamahinga o ngā whārite rārangi me ngā pūnaha whārite rārangi i roto i ngā momo mara, ko ētahi o ēnei ko:
1. Ōhanga
I roto i te ōhanga, ka whakamahia ngā whārite rārangi hei tātari i te taurite i waenga i te tuku me te tono, hei whakatau i ngā utu me ngā rahinga taurite, me te whakatauira i ngā āhuatanga ōhanga rerekē.
2. Te Hangarau me te Ahupūngao
I roto i te hangarau, ka whakamahia ngā whārite rārangi i roto i te tātari ara iahiko hiko, te tātari hanganga me te tātari rauemi, me ētahi atu tono e pā ana ki ngā whanaungatanga taurite i waenga i ngā taurangi ā-tinana.
3. Ngā Pūtaiao Pāpori
He maha ngā wā ka whakamahia ngā whārite rārangi i roto i ngā pūtaiao pāpori hei whakamātautau i ngā whanaungatanga i waenga i ngā taurangi, pērā i te tātari whakatauira i roto i ngā tatauranga.
4. Pūtaiao Rorohiko
Ko ngā rauropi arotau he maha ngā wā ka uru ki te whakaoti rapanga i ngā pūnaha whārite rārangi, hei tauira, i roto i te tātari raraunga, te ako mīhini, me te rangahau mahi.
Whakamutunga
He ariā pāngarau taketake ngā whārite rārangi, ā, he whānuitia ngā whakamahinga. He mea nui te mārama ki te whakaoti rapanga i ngā whārite rārangi me ngā pūnaha whārite rārangi mō ngā mara mai i te ōhanga me te hangarau ki ngā pūtaiao pāpori. Mā te whakamahi i ngā taputapu pēnei i te whakakapinga, te whakakorenga, me te whakamahinga o ngā matihiko, ka taea e tātou te whakaoti i ngā momo raruraru e pā ana ki ngā whanaungatanga rārangi i waenga i ngā taurangi. Mā te mōhio ki ngā whārite rārangi ka huaki te kuaha ki te māramatanga hohonu ake ki te pāngarau me ōna whakamahinga o te ao tūturu.
