Whakaoti i te {l}{x+4y=41}{4x+5y=65}

$$\left. \begin{array} { l } { x + 4 y = 41 } \\ { 4 x + 5 y = 65 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=9$$

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.

$$x+4y=41,4x+5y=65$$

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

$$x+4y=41$$

Me tango 4y mai i ngā taha e rua o te whārite.

$$x=-4y+41$$

Whakakapia te -4y+41 mō te x ki tērā atu whārite, 4x+5y=65.

$$4\left(-4y+41\right)+5y=65$$

Whakareatia 4 ki te -4y+41.

$$-16y+164+5y=65$$

Tāpiri -16y ki te 5y.

$$-11y+164=65$$

Me tango 164 mai i ngā taha e rua o te whārite.

$$-11y=-99$$

Whakawehea ngā taha e rua ki te -11.

$$y=9$$

Whakaurua te 9 mō y ki x=-4y+41. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-4\times 9+41$$

Whakareatia -4 ki te 9.

$$x=-36+41$$

Tāpiri 41 ki te -36.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=9$$

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.

$$x+4y=41,4x+5y=65$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&4\\4&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}41\\65\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&4\\4&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&4\\4&5\end{matrix}\right))\left(\begin{matrix}1&4\\4&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\4&5\end{matrix}\right))\left(\begin{matrix}41\\65\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\4&5\end{matrix}\right))\left(\begin{matrix}41\\65\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\4&5\end{matrix}\right))\left(\begin{matrix}41\\65\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5-4\times 4}&-\frac{4}{5-4\times 4}\\-\frac{4}{5-4\times 4}&\frac{1}{5-4\times 4}\end{matrix}\right)\left(\begin{matrix}41\\65\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{11}&\frac{4}{11}\\\frac{4}{11}&-\frac{1}{11}\end{matrix}\right)\left(\begin{matrix}41\\65\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{11}\times 41+\frac{4}{11}\times 65\\\frac{4}{11}\times 41-\frac{1}{11}\times 65\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\9\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=9$$

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.

$$x+4y=41,4x+5y=65$$

Kia ōrite ai a x 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 1.

$$4x+4\times 4y=4\times 41,4x+5y=65$$

Whakarūnātia.

$$4x+16y=164,4x+5y=65$$

Me tango 4x+5y=65 mai i 4x+16y=164 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-4x+16y-5y=164-65$$

Tāpiri 4x ki te -4x. Ka whakakore atu ngā kupu 4x me -4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$16y-5y=164-65$$

Tāpiri 16y ki te -5y.

$$11y=164-65$$

Tāpiri 164 ki te -65.

$$11y=99$$

Whakawehea ngā taha e rua ki te 11.

$$y=9$$

Whakaurua te 9 mō y ki 4x+5y=65. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+5\times 9=65$$

Whakareatia 5 ki te 9.

$$4x+45=65$$

Me tango 45 mai i ngā taha e rua o te whārite.

$$4x=20$$

Whakawehea ngā taha e rua ki te 4.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=9$$