Whakaoti i te {l}{-9y=13x+81}{1/9y=1+5/81x}

$$\left. \begin{array} { l } { - 9 y = 13 x + 81 } \\ { \frac { 1 } { 9 } y = 1 + \frac { 5 } { 81 } x } \end{array} \right.$$

Whakaoti mō y, x

$$x=-9$$

$$y=4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 13x mai i ngā taha e rua.

$$-9y-13x=81$$

Whakaarohia te whārite tuarua. Tangohia te \frac{5}{81}x mai i ngā taha e rua.

$$\frac{1}{9}y-\frac{5}{81}x=1$$

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.

$$-9y-13x=81,\frac{1}{9}y-\frac{5}{81}x=1$$

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

$$-9y-13x=81$$

Me tāpiri 13x ki ngā taha e rua o te whārite.

$$-9y=13x+81$$

Whakawehea ngā taha e rua ki te -9.

$$y=-\frac{1}{9}\left(13x+81\right)$$

Whakareatia -\frac{1}{9} ki te 13x+81.

$$y=-\frac{13}{9}x-9$$

Whakakapia te -\frac{13x}{9}-9 mō te y ki tērā atu whārite, \frac{1}{9}y-\frac{5}{81}x=1.

$$\frac{1}{9}\left(-\frac{13}{9}x-9\right)-\frac{5}{81}x=1$$

Whakareatia \frac{1}{9} ki te -\frac{13x}{9}-9.

$$-\frac{13}{81}x-1-\frac{5}{81}x=1$$

Tāpiri -\frac{13x}{81} ki te -\frac{5x}{81}.

$$-\frac{2}{9}x-1=1$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$-\frac{2}{9}x=2$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{2}{9}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-9$$

Whakaurua te -9 mō x ki y=-\frac{13}{9}x-9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-\frac{13}{9}\left(-9\right)-9$$

Whakareatia -\frac{13}{9} ki te -9.

$$y=13-9$$

Tāpiri -9 ki te 13.

$$y=4$$

Kua oti te pūnaha te whakatau.

$$y=4,x=-9$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 13x mai i ngā taha e rua.

$$-9y-13x=81$$

Whakaarohia te whārite tuarua. Tangohia te \frac{5}{81}x mai i ngā taha e rua.

$$\frac{1}{9}y-\frac{5}{81}x=1$$

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

$$-9y-13x=81,\frac{1}{9}y-\frac{5}{81}x=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-9&-13\\\frac{1}{9}&-\frac{5}{81}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}81\\1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-9&-13\\\frac{1}{9}&-\frac{5}{81}\end{matrix}\right).

$$inverse(\left(\begin{matrix}-9&-13\\\frac{1}{9}&-\frac{5}{81}\end{matrix}\right))\left(\begin{matrix}-9&-13\\\frac{1}{9}&-\frac{5}{81}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}-9&-13\\\frac{1}{9}&-\frac{5}{81}\end{matrix}\right))\left(\begin{matrix}81\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}-9&-13\\\frac{1}{9}&-\frac{5}{81}\end{matrix}\right))\left(\begin{matrix}81\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}-9&-13\\\frac{1}{9}&-\frac{5}{81}\end{matrix}\right))\left(\begin{matrix}81\\1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{\frac{5}{81}}{-9\left(-\frac{5}{81}\right)-\left(-13\times \frac{1}{9}\right)}&-\frac{-13}{-9\left(-\frac{5}{81}\right)-\left(-13\times \frac{1}{9}\right)}\\-\frac{\frac{1}{9}}{-9\left(-\frac{5}{81}\right)-\left(-13\times \frac{1}{9}\right)}&-\frac{9}{-9\left(-\frac{5}{81}\right)-\left(-13\times \frac{1}{9}\right)}\end{matrix}\right)\left(\begin{matrix}81\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{162}&\frac{13}{2}\\-\frac{1}{18}&-\frac{9}{2}\end{matrix}\right)\left(\begin{matrix}81\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{162}\times 81+\frac{13}{2}\\-\frac{1}{18}\times 81-\frac{9}{2}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}4\\-9\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=4,x=-9$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 13x mai i ngā taha e rua.

$$-9y-13x=81$$

Whakaarohia te whārite tuarua. Tangohia te \frac{5}{81}x mai i ngā taha e rua.

$$\frac{1}{9}y-\frac{5}{81}x=1$$

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.

$$-9y-13x=81,\frac{1}{9}y-\frac{5}{81}x=1$$

Kia ōrite ai a -9y me \frac{y}{9}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te \frac{1}{9} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -9.

$$\frac{1}{9}\left(-9\right)y+\frac{1}{9}\left(-13\right)x=\frac{1}{9}\times 81,-9\times \frac{1}{9}y-9\left(-\frac{5}{81}\right)x=-9$$

Whakarūnātia.

$$-y-\frac{13}{9}x=9,-y+\frac{5}{9}x=-9$$

Me tango -y+\frac{5}{9}x=-9 mai i -y-\frac{13}{9}x=9 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-y+y-\frac{13}{9}x-\frac{5}{9}x=9+9$$

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

$$-\frac{13}{9}x-\frac{5}{9}x=9+9$$

Tāpiri -\frac{13x}{9} ki te -\frac{5x}{9}.

$$-2x=9+9$$

Tāpiri 9 ki te 9.

$$-2x=18$$

Whakawehea ngā taha e rua ki te -2.

$$x=-9$$

Whakaurua te -9 mō x ki \frac{1}{9}y-\frac{5}{81}x=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$\frac{1}{9}y-\frac{5}{81}\left(-9\right)=1$$

Whakareatia -\frac{5}{81} ki te -9.

$$\frac{1}{9}y+\frac{5}{9}=1$$

Me tango \frac{5}{9} mai i ngā taha e rua o te whārite.

$$\frac{1}{9}y=\frac{4}{9}$$

Me whakarea ngā taha e rua ki te 9.

$$y=4$$

Kua oti te pūnaha te whakatau.

$$y=4,x=-9$$