Whakaoti i te {l}{x+y=50}{25x+7y=300}

$$\left\{ \begin{array} { l } { x + y = 50 } \\ { 25 x + 7 y = 300 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{25}{9} = -2\frac{7}{9} \approx -2.777777778$$

$$y = \frac{475}{9} = 52\frac{7}{9} \approx 52.777777778$$

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.

$$x+y=50,25x+7y=300$$

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, 25x+7y=300.

$$25\left(-y+50\right)+7y=300$$

Whakareatia 25 ki te -y+50.

$$-25y+1250+7y=300$$

Tāpiri -25y ki te 7y.

$$-18y+1250=300$$

Me tango 1250 mai i ngā taha e rua o te whārite.

$$-18y=-950$$

Whakawehea ngā taha e rua ki te -18.

$$y=\frac{475}{9}$$

Whakaurua te \frac{475}{9} 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.

$$x=-\frac{475}{9}+50$$

Tāpiri 50 ki te -\frac{475}{9}.

$$x=-\frac{25}{9}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{25}{9},y=\frac{475}{9}$$

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.

$$x+y=50,25x+7y=300$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\25&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\300\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\25&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\25&7\end{matrix}\right))\left(\begin{matrix}1&1\\25&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\25&7\end{matrix}\right))\left(\begin{matrix}50\\300\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\25&7\end{matrix}\right))\left(\begin{matrix}50\\300\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\25&7\end{matrix}\right))\left(\begin{matrix}50\\300\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{7-25}&-\frac{1}{7-25}\\-\frac{25}{7-25}&\frac{1}{7-25}\end{matrix}\right)\left(\begin{matrix}50\\300\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{18}&\frac{1}{18}\\\frac{25}{18}&-\frac{1}{18}\end{matrix}\right)\left(\begin{matrix}50\\300\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{18}\times 50+\frac{1}{18}\times 300\\\frac{25}{18}\times 50-\frac{1}{18}\times 300\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{25}{9}\\\frac{475}{9}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{25}{9},y=\frac{475}{9}$$

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.

$$x+y=50,25x+7y=300$$

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.

$$25x+25y=25\times 50,25x+7y=300$$

Whakarūnātia.

$$25x+25y=1250,25x+7y=300$$

Me tango 25x+7y=300 mai i 25x+25y=1250 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$25x-25x+25y-7y=1250-300$$

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.

$$25y-7y=1250-300$$

Tāpiri 25y ki te -7y.

$$18y=1250-300$$

Tāpiri 1250 ki te -300.

$$18y=950$$

Whakawehea ngā taha e rua ki te 18.

$$y=\frac{475}{9}$$

Whakaurua te \frac{475}{9} mō y ki 25x+7y=300. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$25x+7\times \frac{475}{9}=300$$

Whakareatia 7 ki te \frac{475}{9}.

$$25x+\frac{3325}{9}=300$$

Me tango \frac{3325}{9} mai i ngā taha e rua o te whārite.

$$25x=-\frac{625}{9}$$

Whakawehea ngā taha e rua ki te 25.

$$x=-\frac{25}{9}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{25}{9},y=\frac{475}{9}$$