Tikanga Whakakapinga i roto i ngā Whārite
Pendahuluan
He pūtaiao taketake, he pūtaiao whakahirahira te pāngarau i roto i ngā āhuatanga maha o te ao, mai i ngā pūtaiao taiao ki ngā pūtaiao pāpori. Ko tētahi peka nui o te pāngarau ko te arapūnga, e tūtaki pinepine ana tātou ki ngā whārite maha. Hei whakaoti whārite, ka taea te whakamahi i ngā tikanga me ngā tikanga maha. Ko tētahi tikanga e tino rongonui ana, e whakaakona pinepinetia ana i roto i ngā marautanga mātauranga ko te tikanga whakakapinga.
Ko te tikanga whakakapinga he tikanga mō te whakaoti whārite e uru ana ki te whakakapi i tētahi taurangi ki tētahi whakaaturanga ōrite o tētahi atu taurangi. Mā te mārama me te mahi i te tikanga whakakapinga, ka taea e tātou te whakahaere i ngā raruraru uaua me te kimi i ngā uara taurangi e tutuki ai te whārite. Ka tūhuratia e tēnei tuhinga te tikanga whakakapinga, mai i ngā ariā taketake me ngā mahi whānui ki ngā tauira o tōna whakamahinga i roto i te whakaoti whārite.
Ngā Ariā Taketake o te Tikanga Whakakapinga
I te nuinga o te wā, ko te tikanga whakakapinga he tikanga hei whakaoti i tētahi pūnaha whārite mā te whakakapi i tētahi taurangi i roto i tētahi whārite ki tētahi kīanga ōrite i puta mai i tētahi atu whārite. He tino whai hua tēnei tikanga mō ngā pūnaha whārite rārangi, engari ka taea hoki te whakamahi ki ētahi momo whārite kore-rārangi.
Whakaarohia te pūnaha māmā o ngā whārite rārangi e whai ake nei:
\[
x + y = 8 \quad \text{(1)}
\]
\[
2x – y = 3 \quad \text{(2)}
\]
Ko te taahiraa tuatahi i roto i te tikanga whakakapinga ko te whiriwhiri i tētahi o ngā whārite me te whakaoti mō tētahi o ngā taurangi. Hei tauira, ka taea e tātou te whiriwhiri i te Whārite (1) me te whakaoti mō \( y \):
\[
y = 8 – x \quad \text{(3)}
\]
Ko te taahiraa tuarua, whakakapia te hua mai i te taahiraa tuatahi ki roto i te whārite kē atu. I tēnei wā, ka whakakapia e mātou a \(y\) mai i te Whārite (3) ki roto i te Whārite (2):
\[
2x – (8 – x) = 3
\]
Ko te taahiraa tuatoru, whakaotihia te whārite i puta mai i te whakakapinga:
\[
2x – 8 + x = 3
\]
\[
3x – 8 = 3
\]
\[
3x = 11
\]
\[
x = \frac{11}{3}
\]
Tuawhā o ngā taahiraa, whakakapia te uara o \( x \) kua kitea ki te Whārite (3) hei kimi i te \( y \):
\[
y = 8 – \frac{11}{3}
\]
\[
y = \frac{24}{3} – \frac{11}{3}
\]
\[
y = \frac{13}{3}
\]
Nō reira, ko ngā otinga mō te pūnaha whārite ko \( x = \frac{11}{3} \) me \( y = \frac{13}{3} \).
Ngā Hipanga Whānui i roto i te Tikanga Whakakapinga
Hei whakaoti i tētahi pūnaha whārite mā te whakamahi i te tikanga whakakapinga, ka taea e tātou te whai i ēnei mahi:
1. Kōwhiria tētahi whārite, ka waiho tētahi o ngā taurangi hei kaupapa.
2. Whakakapia te kīanga i whiwhi mai i te taahiraa tuatahi ki roto i te whārite kē atu.
3. Whakaotia te whārite i puta mai i te hua whakakapinga hei kimi i te uara o te taurangi e toe ana.
4. Whakakapia ngā uara taurangi kua kitea ki roto i te whārite taketake hei kimi i ngā uara o ngā taurangi kē atu.
5. Tirohia te otinga mā te mono i ngā uara taurangi ki roto i ngā whārite taketake kia tino tutuki ai ngā whārite e rua.
Ngā Whakamahinga i roto i ngā Momo Whārite Rerekē
Kāore te tikanga whakakapinga e herea noa ki ngā pūnaha whārite rārangi. Ka taea hoki te whakamahi hei whakaoti i ngā whārite kore-rārangi, pērā i ngā whārite tapawhā, ngā whārite taupū, me ngā whārite taupū.
1. Pūnaha o ngā Whārite Tapawhā
Whakaarohia te pūnaha whārite e whai ake nei:
\[
x + y = 5 \quad \text{(1)}
\]
\[
x^2 + y^2 = 25 \quad \text{(2)}
\]
Ka taea e tātou te tīmata mā te whakaoti i te Whārite (1) mō tētahi o ngā taurangi, hei tauira \( y \):
\[
y = 5 – x \quad \text{(3)}
\]
Kātahi, whakakapia te kīanga mai i te Whārite (3) ki te Whārite (2):
\[
x^2 + (5 – x)^2 = 25
\]
\[
x^2 + 25 – 10x + x^2 = 25
\]
\[
2x^2 – 10x + 25 = 25
\]
\[
2x^2 – 10x = 0
\]
\[
2x(x – 5) = 0
\]
Mā te whakaoti i te whārite i runga ake nei ka puta ngā uara e rua \( x \):
\[
x = 0 \quad \text{or} \quad x = 5
\]
Mō \( x = 0 \), whakakapia ki te Whārite (3):
\[
y = 5 – 0
\]
\[
y = 5
\]
Mō \( x = 5 \), whakakapia ki te Whārite (3):
\[
y = 5 – 5
\]
\[
y = 0
\]
Nō reira, ko te otinga mō te pūnaha whārite ko \( (x, y) = (0, 5) \) me \( (x, y) = (5, 0) \).
2. Pūnaha Whārite Taupūnga
Whakaarohia te pūnaha whārite e whai ake nei:
\[
e^x + y = 3 \quad \text{(1)}
\]
\[
e^x – y = 1 \quad \text{(2)}
\]
Ka taea e tātou te tīmata mā te whakaoti i te Whārite (1) mō \( y \):
\[
y = 3 – e^x \quad \text{(3)}
\]
Kātahi ka whakakapia te kīanga mai i te Whārite (3) ki te Whārite (2):
\[
e^x – (3 – e^x) = 1
\]
\[
e^x – 3 + e^x = 1
\]
\[
2e^x = 4
\]
\[
e^x = 2
\]
\[
x = \ln(2)
\]
Whakakapia te uara o \( x = \ln(2) \) ki te Whārite (3):
\[
y = 3 – e^{\ln(2)}
\]
\[
y = 3 – 2
\]
\[
y = 1
\]
Nō reira, ko te otinga mō te pūnaha whārite ko \( x = \ln(2) \) me \( y = 1 \).
Whakamutunga
He taputapu kaha, he taputapu whai hua hoki te tikanga whakakapinga mō te whakaoti rapanga i ngā pūnaha whārite. Mā te mārama me te whakahaere i ngā mahi tika, ka taea e tātou te whakaoti i ngā momo whārite, mai i te whārite rārangi ki te whārite kore-rārangi. Ehara i te mea ka āwhina noa tēnei huarahi ki te whakangawari i ngā pūnaha whārite engari ka whakarato hoki i tētahi turanga pakari mō te mārama ki ngā tikanga whakaoti rapanga uaua ake. Hei whakamutunga, mā te mahi tonu me te whakamahi i tēnei ariā ki ngā momo raruraru ka whakapai ake i ō tātou pūkenga ki te pāngarau me te pāngarau whānui.
