Whakaoti i te {l}{x+y=50}{5x+7y=300}

$$\left\{ \begin{array} { l } { x + y = 50 } \\ { 5 x + 7 y = 300 } \end{array} \right.$$

Whakaoti mō x, y

$$x=25$$

$$y=25$$

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.

$$x+y=50,5x+7y=300$$

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

$$x+y=50$$

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

$$x=-y+50$$

Whakakapia te -y+50 mō te x ki tērā atu whārite, 5x+7y=300.

$$5\left(-y+50\right)+7y=300$$

Whakareatia 5 ki te -y+50.

$$-5y+250+7y=300$$

Tāpiri -5y ki te 7y.

$$2y+250=300$$

Me tango 250 mai i ngā taha e rua o te whārite.

$$2y=50$$

Whakawehea ngā taha e rua ki te 2.

$$y=25$$

Whakaurua te 25 mō y ki x=-y+50. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-25+50$$

Tāpiri 50 ki te -25.

$$x=25$$

Kua oti te pūnaha te whakatau.

$$x=25,y=25$$

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.

$$x+y=50,5x+7y=300$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\5&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\300\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\5&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\5&7\end{matrix}\right))\left(\begin{matrix}1&1\\5&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\5&7\end{matrix}\right))\left(\begin{matrix}50\\300\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\5&7\end{matrix}\right))\left(\begin{matrix}50\\300\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\5&7\end{matrix}\right))\left(\begin{matrix}50\\300\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{7-5}&-\frac{1}{7-5}\\-\frac{5}{7-5}&\frac{1}{7-5}\end{matrix}\right)\left(\begin{matrix}50\\300\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{2}&-\frac{1}{2}\\-\frac{5}{2}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}50\\300\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{2}\times 50-\frac{1}{2}\times 300\\-\frac{5}{2}\times 50+\frac{1}{2}\times 300\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}25\\25\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=25,y=25$$

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.

$$x+y=50,5x+7y=300$$

Kia ōrite ai a x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$5x+5y=5\times 50,5x+7y=300$$

Whakarūnātia.

$$5x+5y=250,5x+7y=300$$

Me tango 5x+7y=300 mai i 5x+5y=250 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5x-5x+5y-7y=250-300$$

Tāpiri 5x ki te -5x. Ka whakakore atu ngā kupu 5x me -5x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$5y-7y=250-300$$

Tāpiri 5y ki te -7y.

$$-2y=250-300$$

Tāpiri 250 ki te -300.

$$-2y=-50$$

Whakawehea ngā taha e rua ki te -2.

$$y=25$$

Whakaurua te 25 mō y ki 5x+7y=300. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x+7\times 25=300$$

Whakareatia 7 ki te 25.

$$5x+175=300$$

Me tango 175 mai i ngā taha e rua o te whārite.

$$5x=125$$

Whakawehea ngā taha e rua ki te 5.

$$x=25$$

Kua oti te pūnaha te whakatau.

$$x=25,y=25$$