Whakaoti i te {l}{7P=B-39}{B-11P=9}

$$\left\{ \begin{array} { l } { 7 P = B - 39 } \\ { B - 11 P = 9 } \end{array} \right.$$

Whakaoti mō P, B

$$P = \frac{15}{2} = 7\frac{1}{2} = 7.5$$

$$B = \frac{183}{2} = 91\frac{1}{2} = 91.5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te B mai i ngā taha e rua.

$$7P-B=-39$$

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$7P-B=-39,-11P+B=9$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te P mā te wehe i te P i te taha mauī o te tohu ōrite.

$$7P-B=-39$$

Me tāpiri B ki ngā taha e rua o te whārite.

$$7P=B-39$$

Whakawehea ngā taha e rua ki te 7.

$$P=\frac{1}{7}\left(B-39\right)$$

Whakareatia \frac{1}{7} ki te B-39.

$$P=\frac{1}{7}B-\frac{39}{7}$$

Whakakapia te \frac{-39+B}{7} mō te P ki tērā atu whārite, -11P+B=9.

$$-11\left(\frac{1}{7}B-\frac{39}{7}\right)+B=9$$

Whakareatia -11 ki te \frac{-39+B}{7}.

$$-\frac{11}{7}B+\frac{429}{7}+B=9$$

Tāpiri -\frac{11B}{7} ki te B.

$$-\frac{4}{7}B+\frac{429}{7}=9$$

Me tango \frac{429}{7} mai i ngā taha e rua o te whārite.

$$-\frac{4}{7}B=-\frac{366}{7}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{4}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$B=\frac{183}{2}$$

Whakaurua te \frac{183}{2} mō B ki P=\frac{1}{7}B-\frac{39}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō P hāngai tonu.

$$P=\frac{1}{7}\times \frac{183}{2}-\frac{39}{7}$$

Whakareatia \frac{1}{7} ki te \frac{183}{2} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$P=\frac{183}{14}-\frac{39}{7}$$

Tāpiri -\frac{39}{7} ki te \frac{183}{14} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$P=\frac{15}{2}$$

Kua oti te pūnaha te whakatau.

$$P=\frac{15}{2},B=\frac{183}{2}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te B mai i ngā taha e rua.

$$7P-B=-39$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$7P-B=-39,-11P+B=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&-1\\-11&1\end{matrix}\right)\left(\begin{matrix}P\\B\end{matrix}\right)=\left(\begin{matrix}-39\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&-1\\-11&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&-1\\-11&1\end{matrix}\right))\left(\begin{matrix}7&-1\\-11&1\end{matrix}\right)\left(\begin{matrix}P\\B\end{matrix}\right)=inverse(\left(\begin{matrix}7&-1\\-11&1\end{matrix}\right))\left(\begin{matrix}-39\\9\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}P\\B\end{matrix}\right)=inverse(\left(\begin{matrix}7&-1\\-11&1\end{matrix}\right))\left(\begin{matrix}-39\\9\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}P\\B\end{matrix}\right)=inverse(\left(\begin{matrix}7&-1\\-11&1\end{matrix}\right))\left(\begin{matrix}-39\\9\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}P\\B\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7-\left(-\left(-11\right)\right)}&-\frac{-1}{7-\left(-\left(-11\right)\right)}\\-\frac{-11}{7-\left(-\left(-11\right)\right)}&\frac{7}{7-\left(-\left(-11\right)\right)}\end{matrix}\right)\left(\begin{matrix}-39\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}P\\B\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&-\frac{1}{4}\\-\frac{11}{4}&-\frac{7}{4}\end{matrix}\right)\left(\begin{matrix}-39\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}P\\B\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\left(-39\right)-\frac{1}{4}\times 9\\-\frac{11}{4}\left(-39\right)-\frac{7}{4}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}P\\B\end{matrix}\right)=\left(\begin{matrix}\frac{15}{2}\\\frac{183}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa P me B.

$$P=\frac{15}{2},B=\frac{183}{2}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te B mai i ngā taha e rua.

$$7P-B=-39$$

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$7P-B=-39,-11P+B=9$$

Kia ōrite ai a 7P me -11P, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -11 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 7.

$$-11\times 7P-11\left(-1\right)B=-11\left(-39\right),7\left(-11\right)P+7B=7\times 9$$

Whakarūnātia.

$$-77P+11B=429,-77P+7B=63$$

Me tango -77P+7B=63 mai i -77P+11B=429 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-77P+77P+11B-7B=429-63$$

Tāpiri -77P ki te 77P. Ka whakakore atu ngā kupu -77P me 77P, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$11B-7B=429-63$$

Tāpiri 11B ki te -7B.

$$4B=429-63$$

Tāpiri 429 ki te -63.

$$4B=366$$

Whakawehea ngā taha e rua ki te 4.

$$B=\frac{183}{2}$$

Whakaurua te \frac{183}{2} mō B ki -11P+B=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō P hāngai tonu.

$$-11P+\frac{183}{2}=9$$

Me tango \frac{183}{2} mai i ngā taha e rua o te whārite.

$$-11P=-\frac{165}{2}$$

Whakawehea ngā taha e rua ki te -11.

$$P=\frac{15}{2}$$

Kua oti te pūnaha te whakatau.

$$P=\frac{15}{2},B=\frac{183}{2}$$