Whakaoti i te {l}{rx-ry=1}{rx-9y=r}

$$\left\{ \begin{array} { l } { r x - r y = 1 } \\ { r x - 9 y = r } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{r^{2}-9}{r\left(r-9\right)}$$

$$y=-\frac{1-r}{r-9}$$

$$r\neq 9\text{ and }r\neq 0$$

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.

$$rx+\left(-r\right)y=1,rx-9y=r$$

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

$$rx+\left(-r\right)y=1$$

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

$$rx=ry+1$$

Whakawehea ngā taha e rua ki te r.

$$x=\frac{1}{r}\left(ry+1\right)$$

Whakareatia \frac{1}{r} ki te ry+1.

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

Whakakapia te y+\frac{1}{r} mō te x ki tērā atu whārite, rx-9y=r.

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

Whakareatia r ki te y+\frac{1}{r}.

$$ry+1-9y=r$$

Tāpiri ry ki te -9y.

$$\left(r-9\right)y+1=r$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$\left(r-9\right)y=r-1$$

Whakawehea ngā taha e rua ki te r-9.

$$y=\frac{r-1}{r-9}$$

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

$$x=\frac{r-1}{r-9}+\frac{1}{r}$$

Tāpiri \frac{1}{r} ki te \frac{r-1}{r-9}.

$$x=\frac{r^{2}-9}{r\left(r-9\right)}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{r^{2}-9}{r\left(r-9\right)},y=\frac{r-1}{r-9}$$

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.

$$rx+\left(-r\right)y=1,rx-9y=r$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{r^{2}-9}{r\left(r-9\right)}\\\frac{r-1}{r-9}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{r^{2}-9}{r\left(r-9\right)},y=\frac{r-1}{r-9}$$

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.

$$rx+\left(-r\right)y=1,rx-9y=r$$

Me tango rx-9y=r mai i rx+\left(-r\right)y=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$rx+\left(-r\right)x+\left(-r\right)y+9y=1-r$$

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

$$\left(-r\right)y+9y=1-r$$

Tāpiri -ry ki te 9y.

$$\left(9-r\right)y=1-r$$

Whakawehea ngā taha e rua ki te -r+9.

$$y=\frac{1-r}{9-r}$$

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

$$rx-9\times \frac{1-r}{9-r}=r$$

Whakareatia -9 ki te \frac{1-r}{-r+9}.

$$rx-\frac{9\left(1-r\right)}{9-r}=r$$

Me tāpiri \frac{9\left(1-r\right)}{-r+9} ki ngā taha e rua o te whārite.

$$rx=-\frac{\left(r-3\right)\left(r+3\right)}{9-r}$$

Whakawehea ngā taha e rua ki te r.

$$x=-\frac{r^{2}-9}{r\left(9-r\right)}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{r^{2}-9}{r\left(9-r\right)},y=\frac{1-r}{9-r}$$