Whakaoti i te {l}{3x-9=y}{9y+3=x}

$$\left. \begin{array} { l } { 3 x - 9 = y } \\ { 9 y + 3 = x } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=0$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$3x-9-y=0$$

Me tāpiri te 9 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3x-y=9$$

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

$$9y+3-x=0$$

Tangohia te 3 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$9y-x=-3$$

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.

$$3x-y=9,-x+9y=-3$$

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

$$3x-y=9$$

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

$$3x=y+9$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(y+9\right)$$

Whakareatia \frac{1}{3} ki te y+9.

$$x=\frac{1}{3}y+3$$

Whakakapia te \frac{y}{3}+3 mō te x ki tērā atu whārite, -x+9y=-3.

$$-\left(\frac{1}{3}y+3\right)+9y=-3$$

Whakareatia -1 ki te \frac{y}{3}+3.

$$-\frac{1}{3}y-3+9y=-3$$

Tāpiri -\frac{y}{3} ki te 9y.

$$\frac{26}{3}y-3=-3$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$\frac{26}{3}y=0$$

Whakawehea ngā taha e rua o te whārite ki te \frac{26}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=0$$

Whakaurua te 0 mō y ki x=\frac{1}{3}y+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=0$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$3x-9-y=0$$

Me tāpiri te 9 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3x-y=9$$

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

$$9y+3-x=0$$

Tangohia te 3 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$9y-x=-3$$

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

$$3x-y=9,-x+9y=-3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-1\\-1&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\-3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-1\\-1&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-1\\-1&9\end{matrix}\right))\left(\begin{matrix}3&-1\\-1&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\-1&9\end{matrix}\right))\left(\begin{matrix}9\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\-1&9\end{matrix}\right))\left(\begin{matrix}9\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\-1&9\end{matrix}\right))\left(\begin{matrix}9\\-3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{3\times 9-\left(-\left(-1\right)\right)}&-\frac{-1}{3\times 9-\left(-\left(-1\right)\right)}\\-\frac{-1}{3\times 9-\left(-\left(-1\right)\right)}&\frac{3}{3\times 9-\left(-\left(-1\right)\right)}\end{matrix}\right)\left(\begin{matrix}9\\-3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{26}&\frac{1}{26}\\\frac{1}{26}&\frac{3}{26}\end{matrix}\right)\left(\begin{matrix}9\\-3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{26}\times 9+\frac{1}{26}\left(-3\right)\\\frac{1}{26}\times 9+\frac{3}{26}\left(-3\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=0$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$3x-9-y=0$$

Me tāpiri te 9 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3x-y=9$$

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

$$9y+3-x=0$$

Tangohia te 3 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$9y-x=-3$$

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.

$$3x-y=9,-x+9y=-3$$

Kia ōrite ai a 3x me -x, 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 3.

$$-3x-\left(-y\right)=-9,3\left(-1\right)x+3\times 9y=3\left(-3\right)$$

Whakarūnātia.

$$-3x+y=-9,-3x+27y=-9$$

Me tango -3x+27y=-9 mai i -3x+y=-9 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-3x+3x+y-27y=-9+9$$

Tāpiri -3x ki te 3x. Ka whakakore atu ngā kupu -3x me 3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$y-27y=-9+9$$

Tāpiri y ki te -27y.

$$-26y=-9+9$$

Tāpiri -9 ki te 9.

$$-26y=0$$

Whakawehea ngā taha e rua ki te -26.

$$y=0$$

Whakaurua te 0 mō y ki -x+9y=-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-x=-3$$

Whakawehea ngā taha e rua ki te -1.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=0$$