Whakaoti i te {l}{x+my=a}{x-ny=b}

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

Whakaoti mō x, y (complex solution)

$$\left\{\begin{matrix}x=\frac{bm+an}{m+n}\text{, }y=-\frac{b-a}{m+n}\text{, }&m\neq -n\\x=ny+b\text{, }y\in \mathrm{C}\text{, }&a=b\text{ and }m=-n\\x=b\text{, }y\in \mathrm{C}\text{, }&m=0\text{ and }n=0\text{ and }a=b\end{matrix}\right.$$

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.

$$x+my=a,x+\left(-n\right)y=b$$

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

$$x+my=a$$

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

$$x=\left(-m\right)y+a$$

Whakakapia te a-my mō te x ki tērā atu whārite, x+\left(-n\right)y=b.

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

Tāpiri -my ki te -ny.

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

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

$$\left(-m-n\right)y=b-a$$

Whakawehea ngā taha e rua ki te -m-n.

$$y=-\frac{b-a}{m+n}$$

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

$$x=\left(-m\right)\left(-\frac{b-a}{m+n}\right)+a$$

Whakareatia -m ki te -\frac{b-a}{m+n}.

$$x=\frac{m\left(b-a\right)}{m+n}+a$$

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

$$x=\frac{bm+an}{m+n}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{bm+an}{m+n},y=-\frac{b-a}{m+n}$$

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.

$$x+my=a,x+\left(-n\right)y=b$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&m\\1&-n\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}a\\b\end{matrix}\right)$$

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{bm+an}{m+n}\\\frac{a-b}{m+n}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{bm+an}{m+n},y=\frac{a-b}{m+n}$$

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.

$$x+my=a,x+\left(-n\right)y=b$$

Me tango x+\left(-n\right)y=b mai i x+my=a mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+my+ny=a-b$$

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

$$my+ny=a-b$$

Tāpiri my ki te ny.

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

Whakawehea ngā taha e rua ki te m+n.

$$y=\frac{a-b}{m+n}$$

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

$$x+\left(-n\right)\times \frac{a-b}{m+n}=b$$

Whakareatia -n ki te \frac{a-b}{m+n}.

$$x-\frac{n\left(a-b\right)}{m+n}=b$$

Me tāpiri \frac{n\left(a-b\right)}{m+n} ki ngā taha e rua o te whārite.

$$x=\frac{bm+an}{m+n}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{bm+an}{m+n},y=\frac{a-b}{m+n}$$

Whakaoti mō x, y

$$\left\{\begin{matrix}x=\frac{bm+an}{m+n}\text{, }y=-\frac{b-a}{m+n}\text{, }&m\neq -n\\x=ny+b\text{, }y\in \mathrm{R}\text{, }&a=b\text{ and }m=-n\\x=b\text{, }y\in \mathrm{R}\text{, }&m=0\text{ and }n=0\text{ and }a=b\end{matrix}\right.$$

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.

$$x+my=a,x+\left(-n\right)y=b$$

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

$$x+my=a$$

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

$$x=\left(-m\right)y+a$$

Whakakapia te a-my mō te x ki tērā atu whārite, x+\left(-n\right)y=b.

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

Tāpiri -my ki te -ny.

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

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

$$\left(-m-n\right)y=b-a$$

Whakawehea ngā taha e rua ki te -m-n.

$$y=-\frac{b-a}{m+n}$$

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

$$x=\left(-m\right)\left(-\frac{b-a}{m+n}\right)+a$$

Whakareatia -m ki te -\frac{b-a}{m+n}.

$$x=\frac{m\left(b-a\right)}{m+n}+a$$

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

$$x=\frac{bm+an}{m+n}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{bm+an}{m+n},y=-\frac{b-a}{m+n}$$

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.

$$x+my=a,x+\left(-n\right)y=b$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&m\\1&-n\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}a\\b\end{matrix}\right)$$

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{bm+an}{m+n}\\\frac{a-b}{m+n}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{bm+an}{m+n},y=\frac{a-b}{m+n}$$

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.

$$x+my=a,x+\left(-n\right)y=b$$

Me tango x+\left(-n\right)y=b mai i x+my=a mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+my+ny=a-b$$

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

$$my+ny=a-b$$

Tāpiri my ki te ny.

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

Whakawehea ngā taha e rua ki te m+n.

$$y=\frac{a-b}{m+n}$$

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

$$x+\left(-n\right)\times \frac{a-b}{m+n}=b$$

Whakareatia -n ki te \frac{a-b}{m+n}.

$$x-\frac{n\left(a-b\right)}{m+n}=b$$

Me tāpiri \frac{n\left(a-b\right)}{m+n} ki ngā taha e rua o te whārite.

$$x=\frac{bm+an}{m+n}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{bm+an}{m+n},y=\frac{a-b}{m+n}$$