Whakaoti i te {l}{2x+9y=19}{4x+my=53}

$$\left\{ \begin{array} { l } { 2 x + 9 y = 19 } \\ { 4 x + m y = 53 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{477-19m}{2\left(m-18\right)}$$

$$y=\frac{15}{m-18}$$

$$m\neq 18$$

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.

$$2x+9y=19,4x+my=53$$

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

$$2x+9y=19$$

Me tango 9y mai i ngā taha e rua o te whārite.

$$2x=-9y+19$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-9y+19\right)$$

Whakareatia \frac{1}{2} ki te -9y+19.

$$x=-\frac{9}{2}y+\frac{19}{2}$$

Whakakapia te \frac{-9y+19}{2} mō te x ki tērā atu whārite, 4x+my=53.

$$4\left(-\frac{9}{2}y+\frac{19}{2}\right)+my=53$$

Whakareatia 4 ki te \frac{-9y+19}{2}.

$$-18y+38+my=53$$

Tāpiri -18y ki te my.

$$\left(m-18\right)y+38=53$$

Me tango 38 mai i ngā taha e rua o te whārite.

$$\left(m-18\right)y=15$$

Whakawehea ngā taha e rua ki te -18+m.

$$y=\frac{15}{m-18}$$

Whakaurua te \frac{15}{-18+m} mō y ki x=-\frac{9}{2}y+\frac{19}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{9}{2}\times \frac{15}{m-18}+\frac{19}{2}$$

Whakareatia -\frac{9}{2} ki te \frac{15}{-18+m}.

$$x=-\frac{135}{2\left(m-18\right)}+\frac{19}{2}$$

Tāpiri \frac{19}{2} ki te -\frac{135}{2\left(-18+m\right)}.

$$x=\frac{19m-477}{2\left(m-18\right)}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{19m-477}{2\left(m-18\right)},y=\frac{15}{m-18}$$

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.

$$2x+9y=19,4x+my=53$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&9\\4&m\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}19\\53\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&9\\4&m\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&9\\4&m\end{matrix}\right))\left(\begin{matrix}2&9\\4&m\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&9\\4&m\end{matrix}\right))\left(\begin{matrix}19\\53\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&9\\4&m\end{matrix}\right))\left(\begin{matrix}19\\53\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&9\\4&m\end{matrix}\right))\left(\begin{matrix}19\\53\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{m}{2m-9\times 4}&-\frac{9}{2m-9\times 4}\\-\frac{4}{2m-9\times 4}&\frac{2}{2m-9\times 4}\end{matrix}\right)\left(\begin{matrix}19\\53\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{m}{2\left(m-18\right)}&-\frac{9}{2\left(m-18\right)}\\-\frac{2}{m-18}&\frac{1}{m-18}\end{matrix}\right)\left(\begin{matrix}19\\53\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{m}{2\left(m-18\right)}\times 19+\left(-\frac{9}{2\left(m-18\right)}\right)\times 53\\\left(-\frac{2}{m-18}\right)\times 19+\frac{1}{m-18}\times 53\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{19m-477}{2\left(m-18\right)}\\\frac{15}{m-18}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{19m-477}{2\left(m-18\right)},y=\frac{15}{m-18}$$

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.

$$2x+9y=19,4x+my=53$$

Kia ōrite ai a 2x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$4\times 2x+4\times 9y=4\times 19,2\times 4x+2my=2\times 53$$

Whakarūnātia.

$$8x+36y=76,8x+2my=106$$

Me tango 8x+2my=106 mai i 8x+36y=76 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+36y+\left(-2m\right)y=76-106$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$36y+\left(-2m\right)y=76-106$$

Tāpiri 36y ki te -2my.

$$\left(36-2m\right)y=76-106$$

Tāpiri 76 ki te -106.

$$\left(36-2m\right)y=-30$$

Whakawehea ngā taha e rua ki te 36-2m.

$$y=-\frac{15}{18-m}$$

Whakaurua te -\frac{15}{18-m} mō y ki 4x+my=53. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+m\left(-\frac{15}{18-m}\right)=53$$

Whakareatia m ki te -\frac{15}{18-m}.

$$4x-\frac{15m}{18-m}=53$$

Me tāpiri \frac{15m}{18-m} ki ngā taha e rua o te whārite.

$$4x=\frac{2\left(477-19m\right)}{18-m}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{477-19m}{2\left(18-m\right)}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{477-19m}{2\left(18-m\right)},y=-\frac{15}{18-m}$$