Whakaoti i te {c}{51=a2^0+c}{13.5=a2^1+c}

$$\left. \begin{array} { c } { 51 = a 2 ^ { 0 } + c } \\ { 13.5 = a 2 ^ { 1 } + c } \end{array} \right.$$

Whakaoti mō a, c

$$a=-37.5$$

$$c=88.5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tātaihia te 2 mā te pū o 0, kia riro ko 1.

$$51=a\times 1+c$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$a\times 1+c=51$$

Whakaraupapatia anō ngā kīanga tau.

$$a+c=51$$

Whakaarohia te whārite tuarua. Tātaihia te 2 mā te pū o 1, kia riro ko 2.

$$13.5=a\times 2+c$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$a\times 2+c=13.5$$

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.

$$a+c=51,2a+c=13.5$$

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

$$a+c=51$$

Me tango c mai i ngā taha e rua o te whārite.

$$a=-c+51$$

Whakakapia te -c+51 mō te a ki tērā atu whārite, 2a+c=13.5.

$$2\left(-c+51\right)+c=13.5$$

Whakareatia 2 ki te -c+51.

$$-2c+102+c=13.5$$

Tāpiri -2c ki te c.

$$-c+102=13.5$$

Me tango 102 mai i ngā taha e rua o te whārite.

$$-c=-88.5$$

Whakawehea ngā taha e rua ki te -1.

$$c=88.5$$

Whakaurua te 88.5 mō c ki a=-c+51. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=-88.5+51$$

Tāpiri 51 ki te -88.5.

$$a=-37.5$$

Kua oti te pūnaha te whakatau.

$$a=-37.5,c=88.5$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tātaihia te 2 mā te pū o 0, kia riro ko 1.

$$51=a\times 1+c$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$a\times 1+c=51$$

Whakaraupapatia anō ngā kīanga tau.

$$a+c=51$$

Whakaarohia te whārite tuarua. Tātaihia te 2 mā te pū o 1, kia riro ko 2.

$$13.5=a\times 2+c$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$a\times 2+c=13.5$$

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

$$a+c=51,2a+c=13.5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\2&1\end{matrix}\right)\left(\begin{matrix}a\\c\end{matrix}\right)=\left(\begin{matrix}51\\13.5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\2&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\2&1\end{matrix}\right))\left(\begin{matrix}1&1\\2&1\end{matrix}\right)\left(\begin{matrix}a\\c\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&1\end{matrix}\right))\left(\begin{matrix}51\\13.5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\c\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&1\end{matrix}\right))\left(\begin{matrix}51\\13.5\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\c\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&1\end{matrix}\right))\left(\begin{matrix}51\\13.5\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\c\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-2}&-\frac{1}{1-2}\\-\frac{2}{1-2}&\frac{1}{1-2}\end{matrix}\right)\left(\begin{matrix}51\\13.5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\c\end{matrix}\right)=\left(\begin{matrix}-1&1\\2&-1\end{matrix}\right)\left(\begin{matrix}51\\13.5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\c\end{matrix}\right)=\left(\begin{matrix}-51+13.5\\2\times 51-13.5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\c\end{matrix}\right)=\left(\begin{matrix}-37.5\\88.5\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me c.

$$a=-37.5,c=88.5$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tātaihia te 2 mā te pū o 0, kia riro ko 1.

$$51=a\times 1+c$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$a\times 1+c=51$$

Whakaraupapatia anō ngā kīanga tau.

$$a+c=51$$

Whakaarohia te whārite tuarua. Tātaihia te 2 mā te pū o 1, kia riro ko 2.

$$13.5=a\times 2+c$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$a\times 2+c=13.5$$

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.

$$a+c=51,2a+c=13.5$$

Me tango 2a+c=13.5 mai i a+c=51 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$a-2a+c-c=51-13.5$$

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

$$a-2a=51-13.5$$

Tāpiri a ki te -2a.

$$-a=51-13.5$$

Tāpiri 51 ki te -13.5.

$$-a=37.5$$

Whakawehea ngā taha e rua ki te -1.

$$a=-37.5$$

Whakaurua te -37.5 mō a ki 2a+c=13.5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō c hāngai tonu.

$$2\left(-37.5\right)+c=13.5$$

Whakareatia 2 ki te -37.5.

$$-75+c=13.5$$

Me tāpiri 75 ki ngā taha e rua o te whārite.

$$c=88.5$$

Kua oti te pūnaha te whakatau.

$$a=-37.5,c=88.5$$