Whakaoti i te {l}{x+y=500}{25x+35y=1450}

$$\left. \begin{array} { l } { x + y = 500 } \\ { 25 x + 35 y = 1450 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1605$$

$$y=-1105$$

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.

$$x+y=500,25x+35y=1450$$

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

$$x+y=500$$

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

$$x=-y+500$$

Whakakapia te -y+500 mō te x ki tērā atu whārite, 25x+35y=1450.

$$25\left(-y+500\right)+35y=1450$$

Whakareatia 25 ki te -y+500.

$$-25y+12500+35y=1450$$

Tāpiri -25y ki te 35y.

$$10y+12500=1450$$

Me tango 12500 mai i ngā taha e rua o te whārite.

$$10y=-11050$$

Whakawehea ngā taha e rua ki te 10.

$$y=-1105$$

Whakaurua te -1105 mō y ki x=-y+500. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\left(-1105\right)+500$$

Whakareatia -1 ki te -1105.

$$x=1105+500$$

Tāpiri 500 ki te 1105.

$$x=1605$$

Kua oti te pūnaha te whakatau.

$$x=1605,y=-1105$$

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.

$$x+y=500,25x+35y=1450$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\25&35\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}500\\1450\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\25&35\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\25&35\end{matrix}\right))\left(\begin{matrix}1&1\\25&35\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\25&35\end{matrix}\right))\left(\begin{matrix}500\\1450\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\25&35\end{matrix}\right))\left(\begin{matrix}500\\1450\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\25&35\end{matrix}\right))\left(\begin{matrix}500\\1450\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{35}{35-25}&-\frac{1}{35-25}\\-\frac{25}{35-25}&\frac{1}{35-25}\end{matrix}\right)\left(\begin{matrix}500\\1450\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{2}&-\frac{1}{10}\\-\frac{5}{2}&\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}500\\1450\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{2}\times 500-\frac{1}{10}\times 1450\\-\frac{5}{2}\times 500+\frac{1}{10}\times 1450\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1605\\-1105\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1605,y=-1105$$

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.

$$x+y=500,25x+35y=1450$$

Kia ōrite ai a x me 25x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 25 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$25x+25y=25\times 500,25x+35y=1450$$

Whakarūnātia.

$$25x+25y=12500,25x+35y=1450$$

Me tango 25x+35y=1450 mai i 25x+25y=12500 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$25x-25x+25y-35y=12500-1450$$

Tāpiri 25x ki te -25x. Ka whakakore atu ngā kupu 25x me -25x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$25y-35y=12500-1450$$

Tāpiri 25y ki te -35y.

$$-10y=12500-1450$$

Tāpiri 12500 ki te -1450.

$$-10y=11050$$

Whakawehea ngā taha e rua ki te -10.

$$y=-1105$$

Whakaurua te -1105 mō y ki 25x+35y=1450. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$25x+35\left(-1105\right)=1450$$

Whakareatia 35 ki te -1105.

$$25x-38675=1450$$

Me tāpiri 38675 ki ngā taha e rua o te whārite.

$$25x=40125$$

Whakawehea ngā taha e rua ki te 25.

$$x=1605$$

Kua oti te pūnaha te whakatau.

$$x=1605,y=-1105$$