Whakaoti i te {l}{5x+y=9}{10x-7y=-18}

$$\left. \begin{array} { l } { 5 x + y = 9 } \\ { 10 x - 7 y = - 18 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=4$$

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.

$$5x+y=9,10x-7y=-18$$

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

$$5x+y=9$$

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

$$5x=-y+9$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-y+9\right)$$

Whakareatia \frac{1}{5} ki te -y+9.

$$x=-\frac{1}{5}y+\frac{9}{5}$$

Whakakapia te \frac{-y+9}{5} mō te x ki tērā atu whārite, 10x-7y=-18.

$$10\left(-\frac{1}{5}y+\frac{9}{5}\right)-7y=-18$$

Whakareatia 10 ki te \frac{-y+9}{5}.

$$-2y+18-7y=-18$$

Tāpiri -2y ki te -7y.

$$-9y+18=-18$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$-9y=-36$$

Whakawehea ngā taha e rua ki te -9.

$$y=4$$

Whakaurua te 4 mō y ki x=-\frac{1}{5}y+\frac{9}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{5}\times 4+\frac{9}{5}$$

Whakareatia -\frac{1}{5} ki te 4.

$$x=\frac{-4+9}{5}$$

Tāpiri \frac{9}{5} ki te -\frac{4}{5} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=4$$

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.

$$5x+y=9,10x-7y=-18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&1\\10&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\-18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&1\\10&-7\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&1\\10&-7\end{matrix}\right))\left(\begin{matrix}5&1\\10&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\10&-7\end{matrix}\right))\left(\begin{matrix}9\\-18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\10&-7\end{matrix}\right))\left(\begin{matrix}9\\-18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\10&-7\end{matrix}\right))\left(\begin{matrix}9\\-18\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{5\left(-7\right)-10}&-\frac{1}{5\left(-7\right)-10}\\-\frac{10}{5\left(-7\right)-10}&\frac{5}{5\left(-7\right)-10}\end{matrix}\right)\left(\begin{matrix}9\\-18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{45}&\frac{1}{45}\\\frac{2}{9}&-\frac{1}{9}\end{matrix}\right)\left(\begin{matrix}9\\-18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{45}\times 9+\frac{1}{45}\left(-18\right)\\\frac{2}{9}\times 9-\frac{1}{9}\left(-18\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=4$$

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.

$$5x+y=9,10x-7y=-18$$

Kia ōrite ai a 5x me 10x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 10 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$10\times 5x+10y=10\times 9,5\times 10x+5\left(-7\right)y=5\left(-18\right)$$

Whakarūnātia.

$$50x+10y=90,50x-35y=-90$$

Me tango 50x-35y=-90 mai i 50x+10y=90 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$50x-50x+10y+35y=90+90$$

Tāpiri 50x ki te -50x. Ka whakakore atu ngā kupu 50x me -50x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$10y+35y=90+90$$

Tāpiri 10y ki te 35y.

$$45y=90+90$$

Tāpiri 90 ki te 90.

$$45y=180$$

Whakawehea ngā taha e rua ki te 45.

$$y=4$$

Whakaurua te 4 mō y ki 10x-7y=-18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$10x-7\times 4=-18$$

Whakareatia -7 ki te 4.

$$10x-28=-18$$

Me tāpiri 28 ki ngā taha e rua o te whārite.

$$10x=10$$

Whakawehea ngā taha e rua ki te 10.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=4$$