Whakaoti i te {l}{f(x)=y+7}{f(y)=9x+8}

$$\left. \begin{array} { l } { f ( x ) = y + 7 } \\ { f ( y ) = 9 x + 8 } \end{array} \right.$$

Whakaoti mō x, y (complex solution)

$$x=\frac{7f+8}{f^{2}-9}$$

$$y=\frac{8f+63}{f^{2}-9}$$

$$f\neq -3\text{ and }f\neq 3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$fx-y=7$$

Whakaarohia te whārite tuarua. Tangohia te 9x mai i ngā taha e rua.

$$fy-9x=8$$

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.

$$fx-y=7,-9x+fy=8$$

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

$$fx-y=7$$

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

$$fx=y+7$$

Whakawehea ngā taha e rua ki te f.

$$x=\frac{1}{f}\left(y+7\right)$$

Whakareatia \frac{1}{f} ki te y+7.

$$x=\frac{1}{f}y+\frac{7}{f}$$

Whakakapia te \frac{7+y}{f} mō te x ki tērā atu whārite, -9x+fy=8.

$$-9\left(\frac{1}{f}y+\frac{7}{f}\right)+fy=8$$

Whakareatia -9 ki te \frac{7+y}{f}.

$$\left(-\frac{9}{f}\right)y-\frac{63}{f}+fy=8$$

Tāpiri -\frac{9y}{f} ki te fy.

$$\left(f-\frac{9}{f}\right)y-\frac{63}{f}=8$$

Me tāpiri \frac{63}{f} ki ngā taha e rua o te whārite.

$$\left(f-\frac{9}{f}\right)y=8+\frac{63}{f}$$

Whakawehea ngā taha e rua ki te f-\frac{9}{f}.

$$y=\frac{8f+63}{f^{2}-9}$$

Whakaurua te \frac{63+8f}{f^{2}-9} mō y ki x=\frac{1}{f}y+\frac{7}{f}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{f}\times \frac{8f+63}{f^{2}-9}+\frac{7}{f}$$

Whakareatia \frac{1}{f} ki te \frac{63+8f}{f^{2}-9}.

$$x=\frac{8f+63}{f\left(f^{2}-9\right)}+\frac{7}{f}$$

Tāpiri \frac{7}{f} ki te \frac{63+8f}{f\left(f^{2}-9\right)}.

$$x=\frac{7f+8}{f^{2}-9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{7f+8}{f^{2}-9},y=\frac{8f+63}{f^{2}-9}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$fx-y=7$$

Whakaarohia te whārite tuarua. Tangohia te 9x mai i ngā taha e rua.

$$fy-9x=8$$

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

$$fx-y=7,-9x+fy=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}f&-1\\-9&f\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\8\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}f&-1\\-9&f\end{matrix}\right))\left(\begin{matrix}f&-1\\-9&f\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}f&-1\\-9&f\end{matrix}\right))\left(\begin{matrix}7\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}f&-1\\-9&f\end{matrix}\right))\left(\begin{matrix}7\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}f&-1\\-9&f\end{matrix}\right))\left(\begin{matrix}7\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{f}{ff-\left(-\left(-9\right)\right)}&-\frac{-1}{ff-\left(-\left(-9\right)\right)}\\-\frac{-9}{ff-\left(-\left(-9\right)\right)}&\frac{f}{ff-\left(-\left(-9\right)\right)}\end{matrix}\right)\left(\begin{matrix}7\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{f}{f^{2}-9}&\frac{1}{f^{2}-9}\\\frac{9}{f^{2}-9}&\frac{f}{f^{2}-9}\end{matrix}\right)\left(\begin{matrix}7\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{f}{f^{2}-9}\times 7+\frac{1}{f^{2}-9}\times 8\\\frac{9}{f^{2}-9}\times 7+\frac{f}{f^{2}-9}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7f+8}{f^{2}-9}\\\frac{8f+63}{f^{2}-9}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{7f+8}{f^{2}-9},y=\frac{8f+63}{f^{2}-9}$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$fx-y=7$$

Whakaarohia te whārite tuarua. Tangohia te 9x mai i ngā taha e rua.

$$fy-9x=8$$

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.

$$fx-y=7,-9x+fy=8$$

Kia ōrite ai a fx me -9x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -9 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te f.

$$-9fx-9\left(-1\right)y=-9\times 7,f\left(-9\right)x+ffy=f\times 8$$

Whakarūnātia.

$$\left(-9f\right)x+9y=-63,\left(-9f\right)x+f^{2}y=8f$$

Me tango \left(-9f\right)x+f^{2}y=8f mai i \left(-9f\right)x+9y=-63 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\left(-9f\right)x+9fx+9y+\left(-f^{2}\right)y=-63-8f$$

Tāpiri -9fx ki te 9fx. Ka whakakore atu ngā kupu -9fx me 9fx, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$9y+\left(-f^{2}\right)y=-63-8f$$

Tāpiri 9y ki te -f^{2}y.

$$\left(9-f^{2}\right)y=-63-8f$$

Tāpiri -63 ki te -8f.

$$\left(9-f^{2}\right)y=-8f-63$$

Whakawehea ngā taha e rua ki te -f^{2}+9.

$$y=-\frac{8f+63}{9-f^{2}}$$

Whakaurua te -\frac{63+8f}{9-f^{2}} mō y ki -9x+fy=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-9x+f\left(-\frac{8f+63}{9-f^{2}}\right)=8$$

Whakareatia f ki te -\frac{63+8f}{9-f^{2}}.

$$-9x-\frac{f\left(8f+63\right)}{9-f^{2}}=8$$

Me tāpiri \frac{f\left(63+8f\right)}{9-f^{2}} ki ngā taha e rua o te whārite.

$$-9x=\frac{9\left(7f+8\right)}{\left(3-f\right)\left(f+3\right)}$$

Whakawehea ngā taha e rua ki te -9.

$$x=-\frac{7f+8}{\left(3-f\right)\left(f+3\right)}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{7f+8}{\left(3-f\right)\left(f+3\right)},y=-\frac{8f+63}{9-f^{2}}$$

Whakaoti mō x, y

$$x=\frac{7f+8}{f^{2}-9}$$

$$y=\frac{8f+63}{f^{2}-9}$$

$$|f|\neq 3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$fx-y=7$$

Whakaarohia te whārite tuarua. Tangohia te 9x mai i ngā taha e rua.

$$fy-9x=8$$

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.

$$fx-y=7,-9x+fy=8$$

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

$$fx-y=7$$

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

$$fx=y+7$$

Whakawehea ngā taha e rua ki te f.

$$x=\frac{1}{f}\left(y+7\right)$$

Whakareatia \frac{1}{f} ki te y+7.

$$x=\frac{1}{f}y+\frac{7}{f}$$

Whakakapia te \frac{7+y}{f} mō te x ki tērā atu whārite, -9x+fy=8.

$$-9\left(\frac{1}{f}y+\frac{7}{f}\right)+fy=8$$

Whakareatia -9 ki te \frac{7+y}{f}.

$$\left(-\frac{9}{f}\right)y-\frac{63}{f}+fy=8$$

Tāpiri -\frac{9y}{f} ki te fy.

$$\left(f-\frac{9}{f}\right)y-\frac{63}{f}=8$$

Me tāpiri \frac{63}{f} ki ngā taha e rua o te whārite.

$$\left(f-\frac{9}{f}\right)y=8+\frac{63}{f}$$

Whakawehea ngā taha e rua ki te f-\frac{9}{f}.

$$y=\frac{8f+63}{f^{2}-9}$$

Whakaurua te \frac{63+8f}{f^{2}-9} mō y ki x=\frac{1}{f}y+\frac{7}{f}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{f}\times \frac{8f+63}{f^{2}-9}+\frac{7}{f}$$

Whakareatia \frac{1}{f} ki te \frac{63+8f}{f^{2}-9}.

$$x=\frac{8f+63}{f\left(f^{2}-9\right)}+\frac{7}{f}$$

Tāpiri \frac{7}{f} ki te \frac{63+8f}{f\left(f^{2}-9\right)}.

$$x=\frac{7f+8}{f^{2}-9}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{7f+8}{f^{2}-9},y=\frac{8f+63}{f^{2}-9}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$fx-y=7$$

Whakaarohia te whārite tuarua. Tangohia te 9x mai i ngā taha e rua.

$$fy-9x=8$$

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

$$fx-y=7,-9x+fy=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}f&-1\\-9&f\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\8\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}f&-1\\-9&f\end{matrix}\right))\left(\begin{matrix}f&-1\\-9&f\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}f&-1\\-9&f\end{matrix}\right))\left(\begin{matrix}7\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}f&-1\\-9&f\end{matrix}\right))\left(\begin{matrix}7\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}f&-1\\-9&f\end{matrix}\right))\left(\begin{matrix}7\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{f}{ff-\left(-\left(-9\right)\right)}&-\frac{-1}{ff-\left(-\left(-9\right)\right)}\\-\frac{-9}{ff-\left(-\left(-9\right)\right)}&\frac{f}{ff-\left(-\left(-9\right)\right)}\end{matrix}\right)\left(\begin{matrix}7\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{f}{f^{2}-9}&\frac{1}{f^{2}-9}\\\frac{9}{f^{2}-9}&\frac{f}{f^{2}-9}\end{matrix}\right)\left(\begin{matrix}7\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{f}{f^{2}-9}\times 7+\frac{1}{f^{2}-9}\times 8\\\frac{9}{f^{2}-9}\times 7+\frac{f}{f^{2}-9}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7f+8}{f^{2}-9}\\\frac{8f+63}{f^{2}-9}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{7f+8}{f^{2}-9},y=\frac{8f+63}{f^{2}-9}$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$fx-y=7$$

Whakaarohia te whārite tuarua. Tangohia te 9x mai i ngā taha e rua.

$$fy-9x=8$$

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.

$$fx-y=7,-9x+fy=8$$

Kia ōrite ai a fx me -9x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -9 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te f.

$$-9fx-9\left(-1\right)y=-9\times 7,f\left(-9\right)x+ffy=f\times 8$$

Whakarūnātia.

$$\left(-9f\right)x+9y=-63,\left(-9f\right)x+f^{2}y=8f$$

Me tango \left(-9f\right)x+f^{2}y=8f mai i \left(-9f\right)x+9y=-63 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\left(-9f\right)x+9fx+9y+\left(-f^{2}\right)y=-63-8f$$

Tāpiri -9fx ki te 9fx. Ka whakakore atu ngā kupu -9fx me 9fx, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$9y+\left(-f^{2}\right)y=-63-8f$$

Tāpiri 9y ki te -f^{2}y.

$$\left(9-f^{2}\right)y=-63-8f$$

Tāpiri -63 ki te -8f.

$$\left(9-f^{2}\right)y=-8f-63$$

Whakawehea ngā taha e rua ki te -f^{2}+9.

$$y=-\frac{8f+63}{9-f^{2}}$$

Whakaurua te -\frac{63+8f}{9-f^{2}} mō y ki -9x+fy=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-9x+f\left(-\frac{8f+63}{9-f^{2}}\right)=8$$

Whakareatia f ki te -\frac{63+8f}{9-f^{2}}.

$$-9x-\frac{f\left(8f+63\right)}{9-f^{2}}=8$$

Me tāpiri \frac{f\left(63+8f\right)}{9-f^{2}} ki ngā taha e rua o te whārite.

$$-9x=\frac{9\left(7f+8\right)}{\left(3-f\right)\left(f+3\right)}$$

Whakawehea ngā taha e rua ki te -9.

$$x=-\frac{7f+8}{\left(3-f\right)\left(f+3\right)}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{7f+8}{\left(3-f\right)\left(f+3\right)},y=-\frac{8f+63}{9-f^{2}}$$