Whakaoti i te {l}{5x+y=7}{-3x+7y=11}

$$\left. \begin{array} { l } { 5 x + y = 7 } \\ { - 3 x + 7 y = 11 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=2$$

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.

$$5x+y=7,-3x+7y=11$$

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

$$5x+y=7$$

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

$$5x=-y+7$$

Whakawehea ngā taha e rua ki te 5.

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

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

$$x=-\frac{1}{5}y+\frac{7}{5}$$

Whakakapia te \frac{-y+7}{5} mō te x ki tērā atu whārite, -3x+7y=11.

$$-3\left(-\frac{1}{5}y+\frac{7}{5}\right)+7y=11$$

Whakareatia -3 ki te \frac{-y+7}{5}.

$$\frac{3}{5}y-\frac{21}{5}+7y=11$$

Tāpiri \frac{3y}{5} ki te 7y.

$$\frac{38}{5}y-\frac{21}{5}=11$$

Me tāpiri \frac{21}{5} ki ngā taha e rua o te whārite.

$$\frac{38}{5}y=\frac{76}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{38}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

Whakaurua te 2 mō y ki x=-\frac{1}{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.

$$x=-\frac{1}{5}\times 2+\frac{7}{5}$$

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

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

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.

$$x=1,y=2$$

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.

$$5x+y=7,-3x+7y=11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&1\\-3&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\11\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&1\\-3&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&1\\-3&7\end{matrix}\right))\left(\begin{matrix}5&1\\-3&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\-3&7\end{matrix}\right))\left(\begin{matrix}7\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\-3&7\end{matrix}\right))\left(\begin{matrix}7\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\-3&7\end{matrix}\right))\left(\begin{matrix}7\\11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{5\times 7-\left(-3\right)}&-\frac{1}{5\times 7-\left(-3\right)}\\-\frac{-3}{5\times 7-\left(-3\right)}&\frac{5}{5\times 7-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}7\\11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{38}&-\frac{1}{38}\\\frac{3}{38}&\frac{5}{38}\end{matrix}\right)\left(\begin{matrix}7\\11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{38}\times 7-\frac{1}{38}\times 11\\\frac{3}{38}\times 7+\frac{5}{38}\times 11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=2$$

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.

$$5x+y=7,-3x+7y=11$$

Kia ōrite ai a 5x me -3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$-3\times 5x-3y=-3\times 7,5\left(-3\right)x+5\times 7y=5\times 11$$

Whakarūnātia.

$$-15x-3y=-21,-15x+35y=55$$

Me tango -15x+35y=55 mai i -15x-3y=-21 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-15x+15x-3y-35y=-21-55$$

Tāpiri -15x ki te 15x. Ka whakakore atu ngā kupu -15x me 15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-3y-35y=-21-55$$

Tāpiri -3y ki te -35y.

$$-38y=-21-55$$

Tāpiri -21 ki te -55.

$$-38y=-76$$

Whakawehea ngā taha e rua ki te -38.

$$y=2$$

Whakaurua te 2 mō y ki -3x+7y=11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-3x+7\times 2=11$$

Whakareatia 7 ki te 2.

$$-3x+14=11$$

Me tango 14 mai i ngā taha e rua o te whārite.

$$-3x=-3$$

Whakawehea ngā taha e rua ki te -3.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=2$$