Whakaoti i te {l}{x=3/4y}{y=8/9x-4}

$$\left\{ \begin{array} { l } { x = \frac { 3 } { 4 } y } \\ { y = \frac { 8 } { 9 } x - 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-9$$

$$y=-12$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te \frac{3}{4}y mai i ngā taha e rua.

$$x-\frac{3}{4}y=0$$

Whakaarohia te whārite tuarua. Tangohia te \frac{8}{9}x mai i ngā taha e rua.

$$y-\frac{8}{9}x=-4$$

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.

$$x-\frac{3}{4}y=0,-\frac{8}{9}x+y=-4$$

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

$$x-\frac{3}{4}y=0$$

Me tāpiri \frac{3y}{4} ki ngā taha e rua o te whārite.

$$x=\frac{3}{4}y$$

Whakakapia te \frac{3y}{4} mō te x ki tērā atu whārite, -\frac{8}{9}x+y=-4.

$$-\frac{8}{9}\times \frac{3}{4}y+y=-4$$

Whakareatia -\frac{8}{9} ki te \frac{3y}{4}.

$$-\frac{2}{3}y+y=-4$$

Tāpiri -\frac{2y}{3} ki te y.

$$\frac{1}{3}y=-4$$

Me whakarea ngā taha e rua ki te 3.

$$y=-12$$

Whakaurua te -12 mō y ki x=\frac{3}{4}y. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{4}\left(-12\right)$$

Whakareatia \frac{3}{4} ki te -12.

$$x=-9$$

Kua oti te pūnaha te whakatau.

$$x=-9,y=-12$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te \frac{3}{4}y mai i ngā taha e rua.

$$x-\frac{3}{4}y=0$$

Whakaarohia te whārite tuarua. Tangohia te \frac{8}{9}x mai i ngā taha e rua.

$$y-\frac{8}{9}x=-4$$

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

$$x-\frac{3}{4}y=0,-\frac{8}{9}x+y=-4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-\frac{3}{4}\\-\frac{8}{9}&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\-4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-\frac{3}{4}\\-\frac{8}{9}&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-\frac{3}{4}\\-\frac{8}{9}&1\end{matrix}\right))\left(\begin{matrix}1&-\frac{3}{4}\\-\frac{8}{9}&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{3}{4}\\-\frac{8}{9}&1\end{matrix}\right))\left(\begin{matrix}0\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{3}{4}\\-\frac{8}{9}&1\end{matrix}\right))\left(\begin{matrix}0\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{3}{4}\\-\frac{8}{9}&1\end{matrix}\right))\left(\begin{matrix}0\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-\left(-\frac{3}{4}\left(-\frac{8}{9}\right)\right)}&-\frac{-\frac{3}{4}}{1-\left(-\frac{3}{4}\left(-\frac{8}{9}\right)\right)}\\-\frac{-\frac{8}{9}}{1-\left(-\frac{3}{4}\left(-\frac{8}{9}\right)\right)}&\frac{1}{1-\left(-\frac{3}{4}\left(-\frac{8}{9}\right)\right)}\end{matrix}\right)\left(\begin{matrix}0\\-4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3&\frac{9}{4}\\\frac{8}{3}&3\end{matrix}\right)\left(\begin{matrix}0\\-4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{4}\left(-4\right)\\3\left(-4\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-9\\-12\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-9,y=-12$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te \frac{3}{4}y mai i ngā taha e rua.

$$x-\frac{3}{4}y=0$$

Whakaarohia te whārite tuarua. Tangohia te \frac{8}{9}x mai i ngā taha e rua.

$$y-\frac{8}{9}x=-4$$

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.

$$x-\frac{3}{4}y=0,-\frac{8}{9}x+y=-4$$

Kia ōrite ai a x me -\frac{8x}{9}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -\frac{8}{9} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$-\frac{8}{9}x-\frac{8}{9}\left(-\frac{3}{4}\right)y=0,-\frac{8}{9}x+y=-4$$

Whakarūnātia.

$$-\frac{8}{9}x+\frac{2}{3}y=0,-\frac{8}{9}x+y=-4$$

Me tango -\frac{8}{9}x+y=-4 mai i -\frac{8}{9}x+\frac{2}{3}y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-\frac{8}{9}x+\frac{8}{9}x+\frac{2}{3}y-y=4$$

Tāpiri -\frac{8x}{9} ki te \frac{8x}{9}. Ka whakakore atu ngā kupu -\frac{8x}{9} me \frac{8x}{9}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\frac{2}{3}y-y=4$$

Tāpiri \frac{2y}{3} ki te -y.

$$-\frac{1}{3}y=4$$

Me whakarea ngā taha e rua ki te -3.

$$y=-12$$

Whakaurua te -12 mō y ki -\frac{8}{9}x+y=-4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-\frac{8}{9}x-12=-4$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$-\frac{8}{9}x=8$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{8}{9}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-9$$

Kua oti te pūnaha te whakatau.

$$x=-9,y=-12$$