Whakaoti i te {l}{8k+a=3650}{15k+a=150}

$$\left\{ \begin{array} { l } { 8 k + a = 3650 } \\ { 15 k + a = 150 } \end{array} \right.$$

Whakaoti mō k, a

$$k=-500$$

$$a=7650$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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.

$$8k+a=3650,15k+a=150$$

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

$$8k+a=3650$$

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

$$8k=-a+3650$$

Whakawehea ngā taha e rua ki te 8.

$$k=\frac{1}{8}\left(-a+3650\right)$$

Whakareatia \frac{1}{8} ki te -a+3650.

$$k=-\frac{1}{8}a+\frac{1825}{4}$$

Whakakapia te -\frac{a}{8}+\frac{1825}{4} mō te k ki tērā atu whārite, 15k+a=150.

$$15\left(-\frac{1}{8}a+\frac{1825}{4}\right)+a=150$$

Whakareatia 15 ki te -\frac{a}{8}+\frac{1825}{4}.

$$-\frac{15}{8}a+\frac{27375}{4}+a=150$$

Tāpiri -\frac{15a}{8} ki te a.

$$-\frac{7}{8}a+\frac{27375}{4}=150$$

Me tango \frac{27375}{4} mai i ngā taha e rua o te whārite.

$$-\frac{7}{8}a=-\frac{26775}{4}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{7}{8}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$a=7650$$

Whakaurua te 7650 mō a ki k=-\frac{1}{8}a+\frac{1825}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō k hāngai tonu.

$$k=-\frac{1}{8}\times 7650+\frac{1825}{4}$$

Whakareatia -\frac{1}{8} ki te 7650.

$$k=\frac{-3825+1825}{4}$$

Tāpiri \frac{1825}{4} ki te -\frac{3825}{4} 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.

$$k=-500$$

Kua oti te pūnaha te whakatau.

$$k=-500,a=7650$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$8k+a=3650,15k+a=150$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}8&1\\15&1\end{matrix}\right)\left(\begin{matrix}k\\a\end{matrix}\right)=\left(\begin{matrix}3650\\150\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}8&1\\15&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&1\\15&1\end{matrix}\right))\left(\begin{matrix}8&1\\15&1\end{matrix}\right)\left(\begin{matrix}k\\a\end{matrix}\right)=inverse(\left(\begin{matrix}8&1\\15&1\end{matrix}\right))\left(\begin{matrix}3650\\150\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}k\\a\end{matrix}\right)=inverse(\left(\begin{matrix}8&1\\15&1\end{matrix}\right))\left(\begin{matrix}3650\\150\end{matrix}\right)$$

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

$$\left(\begin{matrix}k\\a\end{matrix}\right)=inverse(\left(\begin{matrix}8&1\\15&1\end{matrix}\right))\left(\begin{matrix}3650\\150\end{matrix}\right)$$

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.

$$\left(\begin{matrix}k\\a\end{matrix}\right)=\left(\begin{matrix}\frac{1}{8-15}&-\frac{1}{8-15}\\-\frac{15}{8-15}&\frac{8}{8-15}\end{matrix}\right)\left(\begin{matrix}3650\\150\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}k\\a\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{7}&\frac{1}{7}\\\frac{15}{7}&-\frac{8}{7}\end{matrix}\right)\left(\begin{matrix}3650\\150\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}k\\a\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{7}\times 3650+\frac{1}{7}\times 150\\\frac{15}{7}\times 3650-\frac{8}{7}\times 150\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}k\\a\end{matrix}\right)=\left(\begin{matrix}-500\\7650\end{matrix}\right)$$

Tangohia ngā huānga poukapa k me a.

$$k=-500,a=7650$$

Ngā Upane e Whakamahi i te Tangohanga

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.

$$8k+a=3650,15k+a=150$$

Me tango 15k+a=150 mai i 8k+a=3650 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8k-15k+a-a=3650-150$$

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

$$8k-15k=3650-150$$

Tāpiri 8k ki te -15k.

$$-7k=3650-150$$

Tāpiri 3650 ki te -150.

$$-7k=3500$$

Whakawehea ngā taha e rua ki te -7.

$$k=-500$$

Whakaurua te -500 mō k ki 15k+a=150. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$15\left(-500\right)+a=150$$

Whakareatia 15 ki te -500.

$$-7500+a=150$$

Me tāpiri 7500 ki ngā taha e rua o te whārite.

$$a=7650$$

Kua oti te pūnaha te whakatau.

$$k=-500,a=7650$$