Whakaoti i te {l}{50x+y=200}{60x+y=260}

$$\left\{ \begin{array} { l } { 50 x + y = 200 } \\ { 60 x + y = 260 } \end{array} \right.$$

Whakaoti mō x, y

$$x=6$$

$$y=-100$$

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.

$$50x+y=200,60x+y=260$$

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

$$50x+y=200$$

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

$$50x=-y+200$$

Whakawehea ngā taha e rua ki te 50.

$$x=\frac{1}{50}\left(-y+200\right)$$

Whakareatia \frac{1}{50} ki te -y+200.

$$x=-\frac{1}{50}y+4$$

Whakakapia te -\frac{y}{50}+4 mō te x ki tērā atu whārite, 60x+y=260.

$$60\left(-\frac{1}{50}y+4\right)+y=260$$

Whakareatia 60 ki te -\frac{y}{50}+4.

$$-\frac{6}{5}y+240+y=260$$

Tāpiri -\frac{6y}{5} ki te y.

$$-\frac{1}{5}y+240=260$$

Me tango 240 mai i ngā taha e rua o te whārite.

$$-\frac{1}{5}y=20$$

Me whakarea ngā taha e rua ki te -5.

$$y=-100$$

Whakaurua te -100 mō y ki x=-\frac{1}{50}y+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{50}\left(-100\right)+4$$

Whakareatia -\frac{1}{50} ki te -100.

$$x=2+4$$

Tāpiri 4 ki te 2.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=-100$$

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.

$$50x+y=200,60x+y=260$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}50&1\\60&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}200\\260\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}50&1\\60&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}50&1\\60&1\end{matrix}\right))\left(\begin{matrix}50&1\\60&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}50&1\\60&1\end{matrix}\right))\left(\begin{matrix}200\\260\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}50&1\\60&1\end{matrix}\right))\left(\begin{matrix}200\\260\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}50&1\\60&1\end{matrix}\right))\left(\begin{matrix}200\\260\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{50-60}&-\frac{1}{50-60}\\-\frac{60}{50-60}&\frac{50}{50-60}\end{matrix}\right)\left(\begin{matrix}200\\260\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{10}&\frac{1}{10}\\6&-5\end{matrix}\right)\left(\begin{matrix}200\\260\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{10}\times 200+\frac{1}{10}\times 260\\6\times 200-5\times 260\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\-100\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=6,y=-100$$

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.

$$50x+y=200,60x+y=260$$

Me tango 60x+y=260 mai i 50x+y=200 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$50x-60x+y-y=200-260$$

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

$$50x-60x=200-260$$

Tāpiri 50x ki te -60x.

$$-10x=200-260$$

Tāpiri 200 ki te -260.

$$-10x=-60$$

Whakawehea ngā taha e rua ki te -10.

$$x=6$$

Whakaurua te 6 mō x ki 60x+y=260. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$60\times 6+y=260$$

Whakareatia 60 ki te 6.

$$360+y=260$$

Me tango 360 mai i ngā taha e rua o te whārite.

$$y=-100$$

Kua oti te pūnaha te whakatau.

$$x=6,y=-100$$