Whakaoti i te {l}{-4x-2y=-16}{7x-5y=11}

$$\left\{ \begin{array} { l } { - 4 x - 2 y = - 16 } \\ { 7 x - 5 y = 11 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=2$$

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.

$$-4x-2y=-16,7x-5y=11$$

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

$$-4x-2y=-16$$

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

$$-4x=2y-16$$

Whakawehea ngā taha e rua ki te -4.

$$x=-\frac{1}{4}\left(2y-16\right)$$

Whakareatia -\frac{1}{4} ki te -16+2y.

$$x=-\frac{1}{2}y+4$$

Whakakapia te -\frac{y}{2}+4 mō te x ki tērā atu whārite, 7x-5y=11.

$$7\left(-\frac{1}{2}y+4\right)-5y=11$$

Whakareatia 7 ki te -\frac{y}{2}+4.

$$-\frac{7}{2}y+28-5y=11$$

Tāpiri -\frac{7y}{2} ki te -5y.

$$-\frac{17}{2}y+28=11$$

Me tango 28 mai i ngā taha e rua o te whārite.

$$-\frac{17}{2}y=-17$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{17}{2}, 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}{2}y+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times 2+4$$

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

$$x=-1+4$$

Tāpiri 4 ki te -1.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=2$$

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.

$$-4x-2y=-16,7x-5y=11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-4&-2\\7&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-16\\11\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-4&-2\\7&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}-4&-2\\7&-5\end{matrix}\right))\left(\begin{matrix}-4&-2\\7&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&-2\\7&-5\end{matrix}\right))\left(\begin{matrix}-16\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&-2\\7&-5\end{matrix}\right))\left(\begin{matrix}-16\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&-2\\7&-5\end{matrix}\right))\left(\begin{matrix}-16\\11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{-4\left(-5\right)-\left(-2\times 7\right)}&-\frac{-2}{-4\left(-5\right)-\left(-2\times 7\right)}\\-\frac{7}{-4\left(-5\right)-\left(-2\times 7\right)}&-\frac{4}{-4\left(-5\right)-\left(-2\times 7\right)}\end{matrix}\right)\left(\begin{matrix}-16\\11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{34}&\frac{1}{17}\\-\frac{7}{34}&-\frac{2}{17}\end{matrix}\right)\left(\begin{matrix}-16\\11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{34}\left(-16\right)+\frac{1}{17}\times 11\\-\frac{7}{34}\left(-16\right)-\frac{2}{17}\times 11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=2$$

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.

$$-4x-2y=-16,7x-5y=11$$

Kia ōrite ai a -4x me 7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -4.

$$7\left(-4\right)x+7\left(-2\right)y=7\left(-16\right),-4\times 7x-4\left(-5\right)y=-4\times 11$$

Whakarūnātia.

$$-28x-14y=-112,-28x+20y=-44$$

Me tango -28x+20y=-44 mai i -28x-14y=-112 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-28x+28x-14y-20y=-112+44$$

Tāpiri -28x ki te 28x. Ka whakakore atu ngā kupu -28x me 28x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-14y-20y=-112+44$$

Tāpiri -14y ki te -20y.

$$-34y=-112+44$$

Tāpiri -112 ki te 44.

$$-34y=-68$$

Whakawehea ngā taha e rua ki te -34.

$$y=2$$

Whakaurua te 2 mō y ki 7x-5y=11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x-5\times 2=11$$

Whakareatia -5 ki te 2.

$$7x-10=11$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$7x=21$$

Whakawehea ngā taha e rua ki te 7.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=2$$