Whakaoti i te {l}{4x+2y=-18}{-2x-5y=10}

$$\left. \begin{array} { l } { 4 x + 2 y = - 18 } \\ { - 2 x - 5 y = 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{35}{8} = -4\frac{3}{8} = -4.375$$

$$y=-\frac{1}{4}=-0.25$$

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.

$$4x+2y=-18,-2x-5y=10$$

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

$$4x+2y=-18$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$4x=-2y-18$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-2y-18\right)$$

Whakareatia \frac{1}{4} ki te -2y-18.

$$x=-\frac{1}{2}y-\frac{9}{2}$$

Whakakapia te \frac{-y-9}{2} mō te x ki tērā atu whārite, -2x-5y=10.

$$-2\left(-\frac{1}{2}y-\frac{9}{2}\right)-5y=10$$

Whakareatia -2 ki te \frac{-y-9}{2}.

$$y+9-5y=10$$

Tāpiri y ki te -5y.

$$-4y+9=10$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$-4y=1$$

Whakawehea ngā taha e rua ki te -4.

$$y=-\frac{1}{4}$$

Whakaurua te -\frac{1}{4} mō y ki x=-\frac{1}{2}y-\frac{9}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\left(-\frac{1}{4}\right)-\frac{9}{2}$$

Whakareatia -\frac{1}{2} ki te -\frac{1}{4} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{1}{8}-\frac{9}{2}$$

Tāpiri -\frac{9}{2} ki te \frac{1}{8} 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.

$$x=-\frac{35}{8}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{35}{8},y=-\frac{1}{4}$$

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.

$$4x+2y=-18,-2x-5y=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&2\\-2&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-18\\10\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}4&2\\-2&-5\end{matrix}\right))\left(\begin{matrix}4&2\\-2&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\-2&-5\end{matrix}\right))\left(\begin{matrix}-18\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\-2&-5\end{matrix}\right))\left(\begin{matrix}-18\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\-2&-5\end{matrix}\right))\left(\begin{matrix}-18\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{4\left(-5\right)-2\left(-2\right)}&-\frac{2}{4\left(-5\right)-2\left(-2\right)}\\-\frac{-2}{4\left(-5\right)-2\left(-2\right)}&\frac{4}{4\left(-5\right)-2\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}-18\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{16}&\frac{1}{8}\\-\frac{1}{8}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}-18\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{16}\left(-18\right)+\frac{1}{8}\times 10\\-\frac{1}{8}\left(-18\right)-\frac{1}{4}\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{35}{8}\\-\frac{1}{4}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{35}{8},y=-\frac{1}{4}$$

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.

$$4x+2y=-18,-2x-5y=10$$

Kia ōrite ai a 4x 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 4.

$$-2\times 4x-2\times 2y=-2\left(-18\right),4\left(-2\right)x+4\left(-5\right)y=4\times 10$$

Whakarūnātia.

$$-8x-4y=36,-8x-20y=40$$

Me tango -8x-20y=40 mai i -8x-4y=36 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-8x+8x-4y+20y=36-40$$

Tāpiri -8x ki te 8x. Ka whakakore atu ngā kupu -8x me 8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-4y+20y=36-40$$

Tāpiri -4y ki te 20y.

$$16y=36-40$$

Tāpiri 36 ki te -40.

$$16y=-4$$

Whakawehea ngā taha e rua ki te 16.

$$y=-\frac{1}{4}$$

Whakaurua te -\frac{1}{4} mō y ki -2x-5y=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x-5\left(-\frac{1}{4}\right)=10$$

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

$$-2x+\frac{5}{4}=10$$

Me tango \frac{5}{4} mai i ngā taha e rua o te whārite.

$$-2x=\frac{35}{4}$$

Whakawehea ngā taha e rua ki te -2.

$$x=-\frac{35}{8}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{35}{8},y=-\frac{1}{4}$$