Whakaoti i te {l}{5x-2y=7}{2x+7y=-5}

$$\left\{ \begin{array} { l } { 5 x - 2 y = 7 } \\ { 2 x + 7 y = - 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=-1$$

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.

$$5x-2y=7,2x+7y=-5$$

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

$$5x-2y=7$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$5x=2y+7$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(2y+7\right)$$

Whakareatia \frac{1}{5} ki te 2y+7.

$$x=\frac{2}{5}y+\frac{7}{5}$$

Whakakapia te \frac{2y+7}{5} mō te x ki tērā atu whārite, 2x+7y=-5.

$$2\left(\frac{2}{5}y+\frac{7}{5}\right)+7y=-5$$

Whakareatia 2 ki te \frac{2y+7}{5}.

$$\frac{4}{5}y+\frac{14}{5}+7y=-5$$

Tāpiri \frac{4y}{5} ki te 7y.

$$\frac{39}{5}y+\frac{14}{5}=-5$$

Me tango \frac{14}{5} mai i ngā taha e rua o te whārite.

$$\frac{39}{5}y=-\frac{39}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{39}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-1$$

Whakaurua te -1 mō y ki x=\frac{2}{5}y+\frac{7}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{5}\left(-1\right)+\frac{7}{5}$$

Whakareatia \frac{2}{5} ki te -1.

$$x=\frac{-2+7}{5}$$

Tāpiri \frac{7}{5} ki te -\frac{2}{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.

$$x=1,y=-1$$

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.

$$5x-2y=7,2x+7y=-5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-2\\2&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-2\\2&7\end{matrix}\right))\left(\begin{matrix}5&-2\\2&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\2&7\end{matrix}\right))\left(\begin{matrix}7\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\2&7\end{matrix}\right))\left(\begin{matrix}7\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\2&7\end{matrix}\right))\left(\begin{matrix}7\\-5\end{matrix}\right)$$

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{39}&\frac{2}{39}\\-\frac{2}{39}&\frac{5}{39}\end{matrix}\right)\left(\begin{matrix}7\\-5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{39}\times 7+\frac{2}{39}\left(-5\right)\\-\frac{2}{39}\times 7+\frac{5}{39}\left(-5\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=-1$$

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.

$$5x-2y=7,2x+7y=-5$$

Kia ōrite ai a 5x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$2\times 5x+2\left(-2\right)y=2\times 7,5\times 2x+5\times 7y=5\left(-5\right)$$

Whakarūnātia.

$$10x-4y=14,10x+35y=-25$$

Me tango 10x+35y=-25 mai i 10x-4y=14 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x-4y-35y=14+25$$

Tāpiri 10x ki te -10x. Ka whakakore atu ngā kupu 10x me -10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-4y-35y=14+25$$

Tāpiri -4y ki te -35y.

$$-39y=14+25$$

Tāpiri 14 ki te 25.

$$-39y=39$$

Whakawehea ngā taha e rua ki te -39.

$$y=-1$$

Whakaurua te -1 mō y ki 2x+7y=-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+7\left(-1\right)=-5$$

Whakareatia 7 ki te -1.

$$2x-7=-5$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$2x=2$$

Whakawehea ngā taha e rua ki te 2.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-1$$