Whakaoti i te {l}{800=x+20y}{700=x+15y}

$$\left. \begin{array} { l } { 800 = x + 20 y } \\ { 700 = x + 15 y } \end{array} \right.$$

Whakaoti mō x, y

$$x=400$$

$$y=20$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$x+20y=800$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$x+15y=700$$

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.

$$x+20y=800,x+15y=700$$

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

$$x+20y=800$$

Me tango 20y mai i ngā taha e rua o te whārite.

$$x=-20y+800$$

Whakakapia te -20y+800 mō te x ki tērā atu whārite, x+15y=700.

$$-20y+800+15y=700$$

Tāpiri -20y ki te 15y.

$$-5y+800=700$$

Me tango 800 mai i ngā taha e rua o te whārite.

$$-5y=-100$$

Whakawehea ngā taha e rua ki te -5.

$$y=20$$

Whakaurua te 20 mō y ki x=-20y+800. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-20\times 20+800$$

Whakareatia -20 ki te 20.

$$x=-400+800$$

Tāpiri 800 ki te -400.

$$x=400$$

Kua oti te pūnaha te whakatau.

$$x=400,y=20$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$x+20y=800$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$x+15y=700$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$x+20y=800,x+15y=700$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&20\\1&15\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}800\\700\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&20\\1&15\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&20\\1&15\end{matrix}\right))\left(\begin{matrix}1&20\\1&15\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&20\\1&15\end{matrix}\right))\left(\begin{matrix}800\\700\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&20\\1&15\end{matrix}\right))\left(\begin{matrix}800\\700\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&20\\1&15\end{matrix}\right))\left(\begin{matrix}800\\700\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{15-20}&-\frac{20}{15-20}\\-\frac{1}{15-20}&\frac{1}{15-20}\end{matrix}\right)\left(\begin{matrix}800\\700\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3&4\\\frac{1}{5}&-\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}800\\700\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\times 800+4\times 700\\\frac{1}{5}\times 800-\frac{1}{5}\times 700\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}400\\20\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=400,y=20$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$x+20y=800$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$x+15y=700$$

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.

$$x+20y=800,x+15y=700$$

Me tango x+15y=700 mai i x+20y=800 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+20y-15y=800-700$$

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.

$$20y-15y=800-700$$

Tāpiri 20y ki te -15y.

$$5y=800-700$$

Tāpiri 800 ki te -700.

$$5y=100$$

Whakawehea ngā taha e rua ki te 5.

$$y=20$$

Whakaurua te 20 mō y ki x+15y=700. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+15\times 20=700$$

Whakareatia 15 ki te 20.

$$x+300=700$$

Me tango 300 mai i ngā taha e rua o te whārite.

$$x=400$$

Kua oti te pūnaha te whakatau.

$$x=400,y=20$$