Te Whakatau i ngā Whārite Wā Kotahi: He Aratohu Whānui
I roto i te pāngarau, ko te whārite tukutahi, te pūnaha whārite rārangi rānei, he huinga whārite e whakauru ana i te maha ōrite o ngā taurangi. Ko ngā otinga ki ēnei whārite ko ngā uara o ngā taurangi e tutuki ana i ngā whārite katoa i roto i te pūnaha i te wā kotahi. He maha ngā wā ka puta ngā whārite tukutahi i roto i ngā momo marautanga, tae atu ki te ōhanga, te ahupūngao, te matū, me te hangarau. Ka matapakihia e tēnei tuhinga ngā tikanga matua mō te whakaoti whārite tukutahi, mai i te whakakapinga me te whakakorenga ki te whakamahinga o ngā matrix me ngā whakatau.
1. Te Ariā Taketake o ngā Whārite Wā Kotahi
Ko ngā whārite tukutahi e rua, neke atu rānei ngā whārite me ngā taurangi e rua, neke atu rānei. He tauira māmā ko ngā whārite rārangi e rua me ngā taurangi e rua:
\[
\begin{ngā take}
2x + y = 5
3x – y = 4
\end{ngā take}
\]
Ko te whāinga o te whakaoti i tēnei whārite he kimi i ngā uara o \( x \) me \( y \) e tutuki ai ngā whārite e rua.
2. Tikanga Whakakapinga
Ko te tikanga whakakapinga e whai ake nei:
1. Kōwhiria tētahi o ngā whārite ka whakarerekē ki te āhua \( y = \) me te \( x = \ rānei).
2. Whakakapia ngā uara o te whārite tuatahi ki te whārite tuarua.
3. Whakaotia te whārite i puta mai hei kimi i te uara o tētahi taurangi.
4. Whakahokia te uara ki roto i tētahi o ngā whārite taketake hei kimi i te uara o te taurangi kē atu.
Hei tauira, me whakamahi tātou i te tauira o mua.
1. Mai i te whārite tuatahi \( 2x + y = 5 \), ka taea e tātou te whakaatu i \( y \) ki te āhua \( y = 5 – 2x \).
2. Tāpirihia te \( y \) kua kitea ki te whārite tuarua: \( 3x – (5 – 2x) = 4 \).
3. Whakatauhia te \( x \):
\[ 3x – 5 + 2x = 4 \]
\[ 5x – 5 = 4 \]
\[ 5x = 9 \]
\[ x = \frac{9}{5} \]
4. Whakakapia te \( x = \frac{9}{5} \) ki roto i te \( y = 5 – 2x \):
\[ y = 5 – 2\left(\frac{9}{5}\right) = 5 – \frac{18}{5} = 5 – 3.6 = 1.4 \]
Ko ngā uara \( x \) me \( y \) he otinga ki te pūnaha whārite.
3. Tikanga Whakakore
Ko te tikanga whakakore ko te whakakore i tētahi o ngā taurangi mā te tāpiri, te tango rānei i tana whārite tango. Ko ngā mahi:
1. Whakareatia tētahi whārite, e rua rānei, kia ōrite ai te tauwehenga o tētahi o ngā taurangi.
2. Tāpirihia, tangohia rānei ngā whārite e rua hei whakakore i te taurangi.
3. Whakaotia te whārite i puta mai mō te taurangi kotahi.
4. Whakakapia te uara o te taurangi kua whiwhi ki roto i tētahi o ngā whārite taketake hei kimi i te taurangi kē atu.
Me whakamahi tātou i te tauira kotahi hei whakamahi i te tikanga whakakore.
1. Whakareatia te whārite tuatahi ki te 1, me te whārite tuarua ki te 2:
\[
\begin{ngā take}
2x + y = 5
6x – 2y = 8
\end{ngā take}
\]
2. Tāpirihia ngā whārite e rua:
\[
(2x + y) + (6x – 2y) = 5 + 8
\]
\[
8x – y = 13
\]
3. Whakatauhia te \( x \):
\[
8x = 13 + y
Nā te mea kāore tā tātou taahiraa whakakore e whakaputa tika i te \(x\), me whakamātau tātou i tētahi atu taahiraa i roto i te whakakorenga. Hei māmā ake, hei wheako ako hoki, me whakarea ngā taha e rua o te whārite tuatahi mā te tauwehenga o te 2:
Tuatahi,
\[ \rightarrow 4x + 2y = 10 \]
Tuarua, ka taea e tātou te tāpiri atu:
\[ \rightarrow 3x – y = 4 \rightarrow 6x – 2y = 8 \]
I muri i te tāpiri:
\[ (4x + 6x ) + (2y – 2y ) = 10 + 8 \rightarrow 10x =18 \rightarrow x = \frac {18}{10} = 1.8 \]
Whakatauhia mō \(x = 1.8 \):
Kimihia te uara o \( y \):
\[ 2(1.8) + y = 5 \]
\[ 3.6 + y = 5 \rightarrow y = 5 – 3.6 =1.4 \]
Nā te mea kua whakaūtia i roto i ngā tūponotanga e rua, he pumau tā mātou otinga: x= 1.8 me y=1.4
Mā te whakaū ka kite tātou he pumau ngā hua mā te whakakapinga me te whakakorenga.
4. Ngā Matrices me ngā Whakatau
He pai ake tēnei tikanga mō ngā pūnaha he nui ake ngā whārite me ngā taurangi. He tikanga e whakamahia whānuitia ana ngā matihiko me ngā taunga whakatau i roto i te arapūrangi rārangi.
Mena he pūnaha whārite tā tātou pēnei i:
\[
\begin{ngā take}
a_{11}x + a_{12}y = b_1
a_{21}x + a_{22}y = b_2
\end{ngā take}
\]
Ka taea te whakaatu i tēnei whārite i roto i te āhua matihiko:
\[ A \mathbf{x} = \mathbf{b} \]
Kei hea
\[ A = \begin{bmatrix} a_{11} me a_{12} \\ a_{21} me a_{22} \end{bmatrix} \]
\[ \mathbf{x} = \begin{bmatrix} x \\ y \end{bmatrix} \]
\[ \mathbf{b} = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} \]
Mai i konei, ka taea e tātou te tuhi i te otinga mā te whakamahi i te whakamuri o te matihiko:
\[ \mathbf{x} = A^{-1} \mathbf{b} \]
Whakatenatenahia te kaipānui me pēhea te huri i te [wāhi o ngā mōhiotanga taketake]:
Kaiwhakatau o te matihiko:
\[ det(A)= a_{11}\cdot a_{22} – a_{21}\cdot a_{12} \]
dan
\[ A^{-1}= [detA]^{-1} a \]
Hei tauira i te wā e taea ai:
\[
\begin{ngā take}
2x + y = 5
3x – y = 4
\end{ngā take}
\]
Ki:
\[
A=
\begin{bmatrix}
2 me te 1 \ 3 me te -1
\end{bmatrix}
\]
\[
Det (A)= ( 2\cdot -1) – (3\cdot 1)= -2-3=-5, \
\pāngarau{x}=
1/hēkona \begin{bmatrix} -1&-1 \\ -3&2 \end{bmatrix}
=
\begin{bmatrix}
\end{ngā take}
Ko tāku tumanako kua mārama te tuhi i ngā kaupae me pēhea te aromatawai.
Whakamutunga
He taputapu nui ngā whārite tukutahi i roto i te pāngarau me ngā tono o te ao tūturu. He maha ngā huarahi e tukuna ana e ngā tikanga maha—te whakakapinga, te whakakorenga, me ngā matihiko—hei whakaoti i aua mea. Ko te whiriwhiri i te tikanga e whakawhirinaki ana ki te uaua o te pūnaha me te taumata whakamarie o te kaiwhakamahi. He whānui te pāngarau, ā, kaua te maha o ngā tikanga e whakamataku, engari ka whakarato i te whānuitanga o ngā otinga.
