Whakaoti i te {l}{a=x(96/60)}{160-a=x+10(96/60)}

$$\left. \begin{array} { l } { a = x ( \frac { 96 } { 60 } ) } \\ { 160 - a = x + 10 ( \frac { 96 } { 60 } ) } \end{array} \right.$$

Whakaoti mō a, x

$$x = \frac{720}{13} = 55\frac{5}{13} \approx 55.384615385$$

$$a = \frac{1152}{13} = 88\frac{8}{13} \approx 88.615384615$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakahekea te hautanga \frac{96}{60} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 12.

$$a=x\times \frac{8}{5}$$

Tangohia te x\times \frac{8}{5} mai i ngā taha e rua.

$$a-x\times \frac{8}{5}=0$$

Whakareatia te -1 ki te \frac{8}{5}, ka -\frac{8}{5}.

$$a-\frac{8}{5}x=0$$

Whakaarohia te whārite tuarua. Whakahekea te hautanga \frac{96}{60} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 12.

$$160-a=x+10\times \frac{8}{5}$$

Whakareatia te 10 ki te \frac{8}{5}, ka 16.

$$160-a=x+16$$

Tangohia te x mai i ngā taha e rua.

$$160-a-x=16$$

Tangohia te 160 mai i ngā taha e rua.

$$-a-x=16-160$$

Tangohia te 160 i te 16, ka -144.

$$-a-x=-144$$

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.

$$a-\frac{8}{5}x=0,-a-x=-144$$

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

$$a-\frac{8}{5}x=0$$

Me tāpiri \frac{8x}{5} ki ngā taha e rua o te whārite.

$$a=\frac{8}{5}x$$

Whakakapia te \frac{8x}{5} mō te a ki tērā atu whārite, -a-x=-144.

$$-\frac{8}{5}x-x=-144$$

Tāpiri -\frac{8x}{5} ki te -x.

$$-\frac{13}{5}x=-144$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{13}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{720}{13}$$

Whakaurua te \frac{720}{13} mō x ki a=\frac{8}{5}x. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$a=\frac{8}{5}\times \frac{720}{13}$$

Whakareatia \frac{8}{5} ki te \frac{720}{13} 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.

$$a=\frac{1152}{13}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{1152}{13},x=\frac{720}{13}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakahekea te hautanga \frac{96}{60} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 12.

$$a=x\times \frac{8}{5}$$

Tangohia te x\times \frac{8}{5} mai i ngā taha e rua.

$$a-x\times \frac{8}{5}=0$$

Whakareatia te -1 ki te \frac{8}{5}, ka -\frac{8}{5}.

$$a-\frac{8}{5}x=0$$

Whakaarohia te whārite tuarua. Whakahekea te hautanga \frac{96}{60} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 12.

$$160-a=x+10\times \frac{8}{5}$$

Whakareatia te 10 ki te \frac{8}{5}, ka 16.

$$160-a=x+16$$

Tangohia te x mai i ngā taha e rua.

$$160-a-x=16$$

Tangohia te 160 mai i ngā taha e rua.

$$-a-x=16-160$$

Tangohia te 160 i te 16, ka -144.

$$-a-x=-144$$

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

$$a-\frac{8}{5}x=0,-a-x=-144$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-\frac{8}{5}\\-1&-1\end{matrix}\right)\left(\begin{matrix}a\\x\end{matrix}\right)=\left(\begin{matrix}0\\-144\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-\frac{8}{5}\\-1&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-\frac{8}{5}\\-1&-1\end{matrix}\right))\left(\begin{matrix}1&-\frac{8}{5}\\-1&-1\end{matrix}\right)\left(\begin{matrix}a\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{8}{5}\\-1&-1\end{matrix}\right))\left(\begin{matrix}0\\-144\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{8}{5}\\-1&-1\end{matrix}\right))\left(\begin{matrix}0\\-144\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{8}{5}\\-1&-1\end{matrix}\right))\left(\begin{matrix}0\\-144\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-\left(-\frac{8}{5}\left(-1\right)\right)}&-\frac{-\frac{8}{5}}{-1-\left(-\frac{8}{5}\left(-1\right)\right)}\\-\frac{-1}{-1-\left(-\frac{8}{5}\left(-1\right)\right)}&\frac{1}{-1-\left(-\frac{8}{5}\left(-1\right)\right)}\end{matrix}\right)\left(\begin{matrix}0\\-144\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\x\end{matrix}\right)=\left(\begin{matrix}\frac{5}{13}&-\frac{8}{13}\\-\frac{5}{13}&-\frac{5}{13}\end{matrix}\right)\left(\begin{matrix}0\\-144\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}a\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{13}\left(-144\right)\\-\frac{5}{13}\left(-144\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}a\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1152}{13}\\\frac{720}{13}\end{matrix}\right)$$

Tangohia ngā huānga poukapa a me x.

$$a=\frac{1152}{13},x=\frac{720}{13}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakahekea te hautanga \frac{96}{60} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 12.

$$a=x\times \frac{8}{5}$$

Tangohia te x\times \frac{8}{5} mai i ngā taha e rua.

$$a-x\times \frac{8}{5}=0$$

Whakareatia te -1 ki te \frac{8}{5}, ka -\frac{8}{5}.

$$a-\frac{8}{5}x=0$$

Whakaarohia te whārite tuarua. Whakahekea te hautanga \frac{96}{60} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 12.

$$160-a=x+10\times \frac{8}{5}$$

Whakareatia te 10 ki te \frac{8}{5}, ka 16.

$$160-a=x+16$$

Tangohia te x mai i ngā taha e rua.

$$160-a-x=16$$

Tangohia te 160 mai i ngā taha e rua.

$$-a-x=16-160$$

Tangohia te 160 i te 16, ka -144.

$$-a-x=-144$$

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.

$$a-\frac{8}{5}x=0,-a-x=-144$$

Kia ōrite ai a a me -a, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$-a-\left(-\frac{8}{5}x\right)=0,-a-x=-144$$

Whakarūnātia.

$$-a+\frac{8}{5}x=0,-a-x=-144$$

Me tango -a-x=-144 mai i -a+\frac{8}{5}x=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-a+a+\frac{8}{5}x+x=144$$

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.

$$\frac{8}{5}x+x=144$$

Tāpiri \frac{8x}{5} ki te x.

$$\frac{13}{5}x=144$$

Whakawehea ngā taha e rua o te whārite ki te \frac{13}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{720}{13}$$

Whakaurua te \frac{720}{13} mō x ki -a-x=-144. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$-a-\frac{720}{13}=-144$$

Me tāpiri \frac{720}{13} ki ngā taha e rua o te whārite.

$$-a=-\frac{1152}{13}$$

Whakawehea ngā taha e rua ki te -1.

$$a=\frac{1152}{13}$$

Kua oti te pūnaha te whakatau.

$$a=\frac{1152}{13},x=\frac{720}{13}$$