Whakaoti i te {l}{mx-y-3m+1=0}{x+my-3m-1=0}

$$\left. \begin{array} { l } { m x - y - 3 m + 1 = 0 } \\ { x + m y - 3 m - 1 = 0 } \end{array} \right.$$

Whakaoti mō x, y (complex solution)

$$x=\frac{3m^{2}+2m+1}{m^{2}+1}$$

$$y=\frac{3m^{2}-2m+1}{m^{2}+1}$$

$$m\neq -i\text{ and }m\neq i$$

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.

$$mx-y+1-3m=0,x+my-3m-1=0$$

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

$$mx-y+1-3m=0$$

Me tango -3m+1 mai i ngā taha e rua o te whārite.

$$mx-y=3m-1$$

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

$$mx=y+3m-1$$

Whakawehea ngā taha e rua ki te m.

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

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

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

Whakakapia te \frac{y-1+3m}{m} mō te x ki tērā atu whārite, x+my-3m-1=0.

$$\frac{1}{m}y+3-\frac{1}{m}+my-3m-1=0$$

Tāpiri \frac{y}{m} ki te my.

$$\left(m+\frac{1}{m}\right)y+3-\frac{1}{m}-3m-1=0$$

Tāpiri 3-\frac{1}{m} ki te -3m-1.

$$\left(m+\frac{1}{m}\right)y-3m+2-\frac{1}{m}=0$$

Me tango 2-\frac{1}{m}-3m mai i ngā taha e rua o te whārite.

$$\left(m+\frac{1}{m}\right)y=3m-2+\frac{1}{m}$$

Whakawehea ngā taha e rua ki te m+\frac{1}{m}.

$$y=\frac{3m^{2}-2m+1}{m^{2}+1}$$

Whakaurua te \frac{3m^{2}+1-2m}{m^{2}+1} mō y ki x=\frac{1}{m}y+3-\frac{1}{m}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{m}\times \frac{3m^{2}-2m+1}{m^{2}+1}+3-\frac{1}{m}$$

Whakareatia \frac{1}{m} ki te \frac{3m^{2}+1-2m}{m^{2}+1}.

$$x=\frac{3m^{2}-2m+1}{m\left(m^{2}+1\right)}+3-\frac{1}{m}$$

Tāpiri 3-\frac{1}{m} ki te \frac{3m^{2}+1-2m}{m\left(m^{2}+1\right)}.

$$x=\frac{3m^{2}+2m+1}{\left(m-i\right)\left(m+i\right)}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3m^{2}+2m+1}{\left(m-i\right)\left(m+i\right)},y=\frac{3m^{2}-2m+1}{m^{2}+1}$$

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.

$$mx-y+1-3m=0,x+my-3m-1=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{m}{mm-\left(-1\right)}&-\frac{-1}{mm-\left(-1\right)}\\-\frac{1}{mm-\left(-1\right)}&\frac{m}{mm-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}3m-1\\3m+1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{m}{m^{2}+1}&\frac{1}{m^{2}+1}\\-\frac{1}{m^{2}+1}&\frac{m}{m^{2}+1}\end{matrix}\right)\left(\begin{matrix}3m-1\\3m+1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{m}{m^{2}+1}\left(3m-1\right)+\frac{1}{m^{2}+1}\left(3m+1\right)\\\left(-\frac{1}{m^{2}+1}\right)\left(3m-1\right)+\frac{m}{m^{2}+1}\left(3m+1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3m^{2}+2m+1}{m^{2}+1}\\\frac{3m^{2}-2m+1}{m^{2}+1}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{3m^{2}+2m+1}{m^{2}+1},y=\frac{3m^{2}-2m+1}{m^{2}+1}$$

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.

$$mx-y+1-3m=0,x+my-3m-1=0$$

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.

$$mx-y+1-3m=0,mx+mmy+m\left(-3m-1\right)=0$$

Whakarūnātia.

$$mx-y+1-3m=0,mx+m^{2}y-m\left(3m+1\right)=0$$

Me tango mx+m^{2}y-m\left(3m+1\right)=0 mai i mx-y+1-3m=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$mx+\left(-m\right)x-y+\left(-m^{2}\right)y+1-3m+m\left(3m+1\right)=0$$

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.

$$-y+\left(-m^{2}\right)y+1-3m+m\left(3m+1\right)=0$$

Tāpiri -y ki te -m^{2}y.

$$\left(-m^{2}-1\right)y+1-3m+m\left(3m+1\right)=0$$

Tāpiri -3m+1 ki te m\left(3m+1\right).

$$\left(-m^{2}-1\right)y+3m^{2}-2m+1=0$$

Me tango -2m+1+3m^{2} mai i ngā taha e rua o te whārite.

$$\left(-m^{2}-1\right)y=-3m^{2}+2m-1$$

Whakawehea ngā taha e rua ki te -1-m^{2}.

$$y=-\frac{-3m^{2}+2m-1}{m^{2}+1}$$

Whakaurua te -\frac{2m-1-3m^{2}}{1+m^{2}} mō y ki x+my-3m-1=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+m\left(-\frac{-3m^{2}+2m-1}{m^{2}+1}\right)-3m-1=0$$

Whakareatia m ki te -\frac{2m-1-3m^{2}}{1+m^{2}}.

$$x-\frac{m\left(-3m^{2}+2m-1\right)}{m^{2}+1}-3m-1=0$$

Tāpiri -\frac{m\left(2m-1-3m^{2}\right)}{1+m^{2}} ki te -3m-1.

$$x-\frac{3m^{2}+2m+1}{\left(m-i\right)\left(m+i\right)}=0$$

Me tāpiri \frac{2m+3m^{2}+1}{\left(m+i\right)\left(m-i\right)} ki ngā taha e rua o te whārite.

$$x=\frac{3m^{2}+2m+1}{\left(m-i\right)\left(m+i\right)}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3m^{2}+2m+1}{\left(m-i\right)\left(m+i\right)},y=-\frac{-3m^{2}+2m-1}{m^{2}+1}$$

Whakaoti mō x, y

$$x=\frac{3m^{2}+2m+1}{m^{2}+1}$$

$$y=\frac{3m^{2}-2m+1}{m^{2}+1}$$

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.

$$mx-y+1-3m=0,x+my-3m-1=0$$

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

$$mx-y+1-3m=0$$

Me tango -3m+1 mai i ngā taha e rua o te whārite.

$$mx-y=3m-1$$

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

$$mx=y+3m-1$$

Whakawehea ngā taha e rua ki te m.

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

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

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

Whakakapia te \frac{y-1+3m}{m} mō te x ki tērā atu whārite, x+my-3m-1=0.

$$\frac{1}{m}y+3-\frac{1}{m}+my-3m-1=0$$

Tāpiri \frac{y}{m} ki te my.

$$\left(m+\frac{1}{m}\right)y+3-\frac{1}{m}-3m-1=0$$

Tāpiri 3-\frac{1}{m} ki te -3m-1.

$$\left(m+\frac{1}{m}\right)y-3m+2-\frac{1}{m}=0$$

Me tango 2-\frac{1}{m}-3m mai i ngā taha e rua o te whārite.

$$\left(m+\frac{1}{m}\right)y=3m-2+\frac{1}{m}$$

Whakawehea ngā taha e rua ki te m+\frac{1}{m}.

$$y=\frac{3m^{2}-2m+1}{m^{2}+1}$$

Whakaurua te \frac{3m^{2}+1-2m}{m^{2}+1} mō y ki x=\frac{1}{m}y+3-\frac{1}{m}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{m}\times \frac{3m^{2}-2m+1}{m^{2}+1}+3-\frac{1}{m}$$

Whakareatia \frac{1}{m} ki te \frac{3m^{2}+1-2m}{m^{2}+1}.

$$x=\frac{3m^{2}-2m+1}{m\left(m^{2}+1\right)}+3-\frac{1}{m}$$

Tāpiri 3-\frac{1}{m} ki te \frac{3m^{2}+1-2m}{m\left(m^{2}+1\right)}.

$$x=\frac{3m^{2}+2m+1}{m^{2}+1}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3m^{2}+2m+1}{m^{2}+1},y=\frac{3m^{2}-2m+1}{m^{2}+1}$$

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.

$$mx-y+1-3m=0,x+my-3m-1=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{m}{mm-\left(-1\right)}&-\frac{-1}{mm-\left(-1\right)}\\-\frac{1}{mm-\left(-1\right)}&\frac{m}{mm-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}3m-1\\3m+1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{m}{m^{2}+1}&\frac{1}{m^{2}+1}\\-\frac{1}{m^{2}+1}&\frac{m}{m^{2}+1}\end{matrix}\right)\left(\begin{matrix}3m-1\\3m+1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{m}{m^{2}+1}\left(3m-1\right)+\frac{1}{m^{2}+1}\left(3m+1\right)\\\left(-\frac{1}{m^{2}+1}\right)\left(3m-1\right)+\frac{m}{m^{2}+1}\left(3m+1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3m^{2}+2m+1}{m^{2}+1}\\\frac{3m^{2}-2m+1}{m^{2}+1}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{3m^{2}+2m+1}{m^{2}+1},y=\frac{3m^{2}-2m+1}{m^{2}+1}$$

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.

$$mx-y+1-3m=0,x+my-3m-1=0$$

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.

$$mx-y+1-3m=0,mx+mmy+m\left(-3m-1\right)=0$$

Whakarūnātia.

$$mx-y+1-3m=0,mx+m^{2}y-m\left(3m+1\right)=0$$

Me tango mx+m^{2}y-m\left(3m+1\right)=0 mai i mx-y+1-3m=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$mx+\left(-m\right)x-y+\left(-m^{2}\right)y+1-3m+m\left(3m+1\right)=0$$

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.

$$-y+\left(-m^{2}\right)y+1-3m+m\left(3m+1\right)=0$$

Tāpiri -y ki te -m^{2}y.

$$\left(-m^{2}-1\right)y+1-3m+m\left(3m+1\right)=0$$

Tāpiri -3m+1 ki te m\left(3m+1\right).

$$\left(-m^{2}-1\right)y+3m^{2}-2m+1=0$$

Me tango -2m+1+3m^{2} mai i ngā taha e rua o te whārite.

$$\left(-m^{2}-1\right)y=-3m^{2}+2m-1$$

Whakawehea ngā taha e rua ki te -1-m^{2}.

$$y=-\frac{-3m^{2}+2m-1}{m^{2}+1}$$

Whakaurua te -\frac{2m-1-3m^{2}}{1+m^{2}} mō y ki x+my-3m-1=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+m\left(-\frac{-3m^{2}+2m-1}{m^{2}+1}\right)-3m-1=0$$

Whakareatia m ki te -\frac{2m-1-3m^{2}}{1+m^{2}}.

$$x-\frac{m\left(-3m^{2}+2m-1\right)}{m^{2}+1}-3m-1=0$$

Tāpiri -\frac{m\left(2m-1-3m^{2}\right)}{1+m^{2}} ki te -3m-1.

$$x-\frac{3m^{2}+2m+1}{m^{2}+1}=0$$

Me tāpiri \frac{2m+3m^{2}+1}{1+m^{2}} ki ngā taha e rua o te whārite.

$$x=\frac{3m^{2}+2m+1}{m^{2}+1}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3m^{2}+2m+1}{m^{2}+1},y=-\frac{-3m^{2}+2m-1}{m^{2}+1}$$