Ngā Tauira Pātai e Matapaki ana i ngā Pūnaha Tauritekore Rārangi
Ko te pūnaha o ngā taurite kore rārangi he peka o te pāngarau e uru ana ki ngā whanaungatanga i waenga i ētahi taurite kore rārangi. Kei roto i tēnei pūnaha e rua, neke atu rānei ngā taurite kore e hiahiatia ana te whakaoti hei kimi i tētahi huinga otinga e ea ai ngā taurite kore katoa i te wā kotahi. He maha ngā wā ka tūtakihia ngā kōrero mō ngā pūnaha taurite kore rārangi i roto i te marautanga pāngarau i ngā taumata kura tuarua me ngā kura tuarua, i roto i ngā pātai whakamātautau me ngā mahi o ia rā.
He maha ngā whakamahinga o ngā pūnaha taurite kore rārangi i roto i te ao tūturu, mai i te arotau rauemi me te whakamahere pūtea ki ngā mahi whakahaere. Ehara i te mea he mea nui anake te mārama ki ēnei ariā mō te whakaoti rapanga pāngarau i te kura, engari he mea whakarite hoki i ngā ākonga ki te whakaoti rapanga o ia rā i runga i te arorau me te whai hua. Kei raro nei ētahi tauira rapanga me ngā kōrero mō ngā pūnaha taurite kore rārangi.
Tauira Pātai 1
Pātai:
Whakatauhia te huinga otinga o te pūnaha taurite kore e whai ake nei:
\[
\begin{ngā take}
x + y \leq 6 \\
x – y \geq 2
\end{ngā take}
\]
Kōrero:
1. Tuhia he rārangi rohe mō ia taurite kore:
Mō \(x + y \leq 6\), ka tuhia e mātou te rārangi \(x + y = 6\):
– Ina puta te pūwāhi (0, 6) i te \(x = 0\), ka puta te pūwāhi (0, 6) i te \(y = 6\).
– Ina puta te pūwāhi (6, 0) i te \(y = 0\), ka puta te pūwāhi (6, 0) i te \(x = 6\).
Mō \(x – y \geq 2\), ka tuhia e mātou te rārangi \(x – y = 2\):
– Ina puta te pūwāhi (2, 0) i te \(x = 2\), ka puta te pūwāhi (0, 0) i te \(y = 6\).
– Ina puta te pūwāhi (0, -2) i te \(y = -2\), ka puta te pūwāhi (0, -2) i te \(x = 0\).
2. Whakatauhia te rohe nohoanga:
– Ka wehea e te rārangi \(x + y = 6\) kia rua ngā rohe, ā, ka tirohia e mātou tētahi pūwāhi whakamātautau kāore i te rārangi, hei tauira te pūwāhi (0, 0):
\[
0 + 0 \leq 6 \quad (\text{true})
\]
Nō reira, ko te horahanga e tutuki ana kei raro iho, kei te taha maui rānei o te rārangi \(x + y = 6\).
– Ka wehea anō te mata e te rārangi \(x – y = 2\) kia rua ngā rohe, ā, ka tirohia e tātou te pūwāhi (0, 0):
\[
0 – 0 \geq 2 \quad (\text{false})
\]
Nō reira, ko te horahanga e tutuki ana kei runga ake, kei te taha matau rānei o te rārangi \(x – y = 2\).
3. Whakatauhia te hononga o ngā rohe e rua:
Ko te otinga mō te pūnaha ko te rohe e tutuki ana i ngā taurite kore e rua. Ka rapua e mātou te hononga o ngā rohe e rua e rite ana ki te ahunga o ia taurite kore.
Ngā tauira:
Ko te huinga otinga o tētahi pūnaha o ngā taurite kore rārangi ko ngā pūwāhi katoa i te whakawhitinga o ngā rohe e rua e tutuki ana i ngā tikanga \(x + y \leq 6\) me \(x – y \geq 2\).
Tauira Pātai 2
Pātai:
Whakatauhia te huinga otinga o te pūnaha taurite kore rārangi e whai ake nei i te hauwhā tuatahi:
\[
\begin{ngā take}
2x + 3y \leq 12 \\
x \geq 0 \\
y \geq 0 \\
\end{ngā take}
\]
Kōrero:
1. Tuhia he rārangi rohe mō ia taurite kore:
Mō \(2x + 3y \leq 12\), ka tuhia e tātou te rārangi \(2x + 3y = 12\):
– Ina puta te pūwāhi (0, 4) i te \(x = 0\), ka puta te pūwāhi (0, 4) i te \(y = 6\).
– Ina puta te pūwāhi (6, 0) i te \(y = 0\), ka puta te pūwāhi (6, 0) i te \(x = 6\).
2. Whakatauhia te rohe nohoanga:
– Rārangi \(2x + 3y = 12\) me te pūwāhi whakamātautau (0, 0):
\[
2(0) + 3(0) \leq 12 \quad (\text{true})
\]
Nō reira, ko te horahanga e tutuki ana kei raro iho, kei te taha maui rānei o te rārangi \(2x + 3y = 12\).
– E tohu ana a \(x \geq 0\) me \(y \geq 0\) kei te hauwhā tuatahi te pūwāhi otinga.
3. Whakatauhia te hononga o ngā rohe e rua:
Ko te otinga mō te pūnaha ko te horahanga i te hauwhā tuatahi kei raro iho, kei te taha maui rānei o te rārangi \(2x + 3y = 12\).
Ngā tauira:
Ko te huinga otinga o tētahi pūnaha o ngā taurite kore rārangi ko ngā pūwāhi i te hauwhā tuatahi e tutuki ana i te \(2x + 3y \leq 12\).
Tauira Pātai 3
Pātai:
Whakatauhia te huinga otinga o te pūnaha taurite kore e whai ake nei:
\[
\begin{ngā take}
y \geq 2x – 3 \\
y \leq -x + 1
\end{ngā take}
\]
Kōrero:
1. Tuhia he rārangi rohe mō ia taurite kore:
Mō \(y \geq 2x – 3\), ka tuhia e tātou te rārangi \(y = 2x – 3\):
– Ina puta te pūwāhi (0, -3) i te \(x = 0\), ka puta te pūwāhi (0, -3) i te \(y = -3\).
– Ina puta te pūwāhi (1,5, 0) i te \(y = 0\), ka puta te pūwāhi (1.5, 0) i te \(x = 6\).
Mō \(y \leq -x + 1\), ka tuhia e mātou te rārangi \(y = -x + 1\):
– Ina puta te pūwāhi (0, 1) i te \(x = 0\), ka puta te pūwāhi (0, 1) i te \(y = 6\).
– Ina puta te pūwāhi (1, 0) i te \(y = 0\), ka puta te pūwāhi (1, 0) i te \(x = 6\).
2. Whakatauhia te rohe nohoanga:
– Ka whakamatautauria te rārangi \(y \geq 2x – 3\) ki te pūwāhi (0, 0):
\[
0 \geq 2(0) – 3 \quad (\text{true})
\]
Nō reira, ko te horahanga e tutuki ana kei runga ake, kei te taha matau rānei o te rārangi \(2x – 3\).
– Ka whakamatautauria te rārangi \(y \leq -x + 1\) ki te pūwāhi (0, 0):
\[
0 \leq -0 + 1 \quad (\text{true})
\]
Nō reira, ko te horahanga e tutuki ana kei raro iho, kei te taha maui rānei o te rārangi \(-x + 1\).
3. Whakatauhia te hononga o ngā rohe e rua:
Ko te otinga mō te pūnaha ko te rohe e tutuki ana i ngā taurite kore e rua. Kei te rapu tātou i te rohe e tūtaki ai ngā taurite kore e rua.
Ngā tauira:
Ko te huinga otinga o tētahi pūnaha o ngā taurite kore rārangi ko ngā pūwāhi i te whakawhitinga o te rohe e tutuki ana i a \(y \geq 2x – 3\) me \(y \leq -x + 1\).
Mā te mārama ki te whakaoti rapanga pāngarau, e tūmanakohia ana ka nui ake te pūkenga o ngā ākonga ki te whakaoti rapanga pāngarau me te whakamahi i ēnei ariā ki ngā āhuatanga o ia rā. Ko te tūmanako, mā ēnei tauira rapanga me ngā kōrerorero ka āwhina i ngā ākonga ki te ako me te mārama ki ngā ariā taketake o ngā pūnaha pāngarau.
