Whakaoti i te {l}{mx-ny=m^2+n^2}{x+y=2m}

$$\left. \begin{array} { l } { m x - n y = m ^ { 2 } + n ^ { 2 } } \\ { x + y = 2 m } \end{array} \right.$$

Whakaoti mō x, y (complex solution)

$$\left\{\begin{matrix}\\x=m+n\text{, }y=m-n\text{, }&\text{unconditionally}\\x=2m-y\text{, }y\in \mathrm{C}\text{, }&m=-n\end{matrix}\right.$$

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.

$$mx+\left(-n\right)y=m^{2}+n^{2},x+y=2m$$

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

$$mx+\left(-n\right)y=m^{2}+n^{2}$$

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

$$mx=ny+m^{2}+n^{2}$$

Whakawehea ngā taha e rua ki te m.

$$x=\frac{1}{m}\left(ny+m^{2}+n^{2}\right)$$

Whakareatia \frac{1}{m} ki te ny+m^{2}+n^{2}.

$$x=\frac{n}{m}y+\frac{n^{2}}{m}+m$$

Whakakapia te \frac{m^{2}+ny+n^{2}}{m} mō te x ki tērā atu whārite, x+y=2m.

$$\frac{n}{m}y+\frac{n^{2}}{m}+m+y=2m$$

Tāpiri \frac{ny}{m} ki te y.

$$\frac{m+n}{m}y+\frac{n^{2}}{m}+m=2m$$

Me tango m+\frac{n^{2}}{m} mai i ngā taha e rua o te whārite.

$$\frac{m+n}{m}y=-\frac{n^{2}}{m}+m$$

Whakawehea ngā taha e rua ki te \frac{m+n}{m}.

$$y=m-n$$

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

$$x=\frac{n}{m}\left(m-n\right)+\frac{n^{2}}{m}+m$$

Whakareatia \frac{n}{m} ki te m-n.

$$x=\frac{n\left(m-n\right)}{m}+\frac{n^{2}}{m}+m$$

Tāpiri m+\frac{n^{2}}{m} ki te \frac{n\left(m-n\right)}{m}.

$$x=m+n$$

Kua oti te pūnaha te whakatau.

$$x=m+n,y=m-n$$

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.

$$mx+\left(-n\right)y=m^{2}+n^{2},x+y=2m$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}m&-n\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}m^{2}+n^{2}\\2m\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}m&-n\\1&1\end{matrix}\right))\left(\begin{matrix}m&-n\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}m&-n\\1&1\end{matrix}\right))\left(\begin{matrix}m^{2}+n^{2}\\2m\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}m&-n\\1&1\end{matrix}\right))\left(\begin{matrix}m^{2}+n^{2}\\2m\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}m&-n\\1&1\end{matrix}\right))\left(\begin{matrix}m^{2}+n^{2}\\2m\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{m-\left(-n\right)}&-\frac{-n}{m-\left(-n\right)}\\-\frac{1}{m-\left(-n\right)}&\frac{m}{m-\left(-n\right)}\end{matrix}\right)\left(\begin{matrix}m^{2}+n^{2}\\2m\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{m+n}&\frac{n}{m+n}\\-\frac{1}{m+n}&\frac{m}{m+n}\end{matrix}\right)\left(\begin{matrix}m^{2}+n^{2}\\2m\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{m+n}\left(m^{2}+n^{2}\right)+\frac{n}{m+n}\times 2m\\\left(-\frac{1}{m+n}\right)\left(m^{2}+n^{2}\right)+\frac{m}{m+n}\times 2m\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}m+n\\m-n\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=m+n,y=m-n$$

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.

$$mx+\left(-n\right)y=m^{2}+n^{2},x+y=2m$$

Kia ōrite ai a mx 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 m.

$$mx+\left(-n\right)y=m^{2}+n^{2},mx+my=m\times 2m$$

Whakarūnātia.

$$mx+\left(-n\right)y=m^{2}+n^{2},mx+my=2m^{2}$$

Me tango mx+my=2m^{2} mai i mx+\left(-n\right)y=m^{2}+n^{2} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$mx+\left(-m\right)x+\left(-n\right)y+\left(-m\right)y=m^{2}+n^{2}-2m^{2}$$

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

$$\left(-n\right)y+\left(-m\right)y=m^{2}+n^{2}-2m^{2}$$

Tāpiri -ny ki te -my.

$$\left(-m-n\right)y=m^{2}+n^{2}-2m^{2}$$

Tāpiri m^{2}+n^{2} ki te -2m^{2}.

$$\left(-m-n\right)y=\left(n-m\right)\left(m+n\right)$$

Whakawehea ngā taha e rua ki te -m-n.

$$y=m-n$$

Whakaurua te m-n mō y ki x+y=2m. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+m-n=2m$$

Me tango m-n mai i ngā taha e rua o te whārite.

$$x=m+n$$

Kua oti te pūnaha te whakatau.

$$x=m+n,y=m-n$$

Whakaoti mō x, y

$$\left\{\begin{matrix}\\x=m+n\text{, }y=m-n\text{, }&\text{unconditionally}\\x=2m-y\text{, }y\in \mathrm{R}\text{, }&m=-n\end{matrix}\right.$$

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.

$$mx+\left(-n\right)y=m^{2}+n^{2},x+y=2m$$

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

$$mx+\left(-n\right)y=m^{2}+n^{2}$$

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

$$mx=ny+m^{2}+n^{2}$$

Whakawehea ngā taha e rua ki te m.

$$x=\frac{1}{m}\left(ny+m^{2}+n^{2}\right)$$

Whakareatia \frac{1}{m} ki te ny+m^{2}+n^{2}.

$$x=\frac{n}{m}y+\frac{n^{2}}{m}+m$$

Whakakapia te \frac{m^{2}+ny+n^{2}}{m} mō te x ki tērā atu whārite, x+y=2m.

$$\frac{n}{m}y+\frac{n^{2}}{m}+m+y=2m$$

Tāpiri \frac{ny}{m} ki te y.

$$\frac{m+n}{m}y+\frac{n^{2}}{m}+m=2m$$

Me tango m+\frac{n^{2}}{m} mai i ngā taha e rua o te whārite.

$$\frac{m+n}{m}y=-\frac{n^{2}}{m}+m$$

Whakawehea ngā taha e rua ki te \frac{m+n}{m}.

$$y=m-n$$

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

$$x=\frac{n}{m}\left(m-n\right)+\frac{n^{2}}{m}+m$$

Whakareatia \frac{n}{m} ki te m-n.

$$x=\frac{n\left(m-n\right)}{m}+\frac{n^{2}}{m}+m$$

Tāpiri m+\frac{n^{2}}{m} ki te \frac{n\left(m-n\right)}{m}.

$$x=m+n$$

Kua oti te pūnaha te whakatau.

$$x=m+n,y=m-n$$

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.

$$mx+\left(-n\right)y=m^{2}+n^{2},x+y=2m$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}m&-n\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}m^{2}+n^{2}\\2m\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}m&-n\\1&1\end{matrix}\right))\left(\begin{matrix}m&-n\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}m&-n\\1&1\end{matrix}\right))\left(\begin{matrix}m^{2}+n^{2}\\2m\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}m&-n\\1&1\end{matrix}\right))\left(\begin{matrix}m^{2}+n^{2}\\2m\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}m&-n\\1&1\end{matrix}\right))\left(\begin{matrix}m^{2}+n^{2}\\2m\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{m-\left(-n\right)}&-\frac{-n}{m-\left(-n\right)}\\-\frac{1}{m-\left(-n\right)}&\frac{m}{m-\left(-n\right)}\end{matrix}\right)\left(\begin{matrix}m^{2}+n^{2}\\2m\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{m+n}&\frac{n}{m+n}\\-\frac{1}{m+n}&\frac{m}{m+n}\end{matrix}\right)\left(\begin{matrix}m^{2}+n^{2}\\2m\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{m+n}\left(m^{2}+n^{2}\right)+\frac{n}{m+n}\times 2m\\\left(-\frac{1}{m+n}\right)\left(m^{2}+n^{2}\right)+\frac{m}{m+n}\times 2m\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}m+n\\m-n\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=m+n,y=m-n$$

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.

$$mx+\left(-n\right)y=m^{2}+n^{2},x+y=2m$$

Kia ōrite ai a mx 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 m.

$$mx+\left(-n\right)y=m^{2}+n^{2},mx+my=m\times 2m$$

Whakarūnātia.

$$mx+\left(-n\right)y=m^{2}+n^{2},mx+my=2m^{2}$$

Me tango mx+my=2m^{2} mai i mx+\left(-n\right)y=m^{2}+n^{2} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$mx+\left(-m\right)x+\left(-n\right)y+\left(-m\right)y=m^{2}+n^{2}-2m^{2}$$

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

$$\left(-n\right)y+\left(-m\right)y=m^{2}+n^{2}-2m^{2}$$

Tāpiri -ny ki te -my.

$$\left(-m-n\right)y=m^{2}+n^{2}-2m^{2}$$

Tāpiri m^{2}+n^{2} ki te -2m^{2}.

$$\left(-m-n\right)y=\left(n-m\right)\left(m+n\right)$$

Whakawehea ngā taha e rua ki te -m-n.

$$y=m-n$$

Whakaurua te m-n mō y ki x+y=2m. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+m-n=2m$$

Me tango m-n mai i ngā taha e rua o te whārite.

$$x=m+n$$

Kua oti te pūnaha te whakatau.

$$x=m+n,y=m-n$$