Whakaoti i te {l}{4x+2y=18}{-3x-6y=27}

$$\left. \begin{array} { l } { 4 x + 2 y = 18 } \\ { - 3 x - 6 y = 27 } \end{array} \right.$$

Whakaoti mō x, y

$$x=9$$

$$y=-9$$

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.

$$4x+2y=18,-3x-6y=27$$

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, -3x-6y=27.

$$-3\left(-\frac{1}{2}y+\frac{9}{2}\right)-6y=27$$

Whakareatia -3 ki te \frac{-y+9}{2}.

$$\frac{3}{2}y-\frac{27}{2}-6y=27$$

Tāpiri \frac{3y}{2} ki te -6y.

$$-\frac{9}{2}y-\frac{27}{2}=27$$

Me tāpiri \frac{27}{2} ki ngā taha e rua o te whārite.

$$-\frac{9}{2}y=\frac{81}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{9}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-9$$

Whakaurua te -9 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.

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

Whakareatia -\frac{1}{2} ki te -9.

$$x=\frac{9+9}{2}$$

Tāpiri \frac{9}{2} ki te \frac{9}{2} 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.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=-9$$

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.

$$4x+2y=18,-3x-6y=27$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&2\\-3&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}18\\27\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&2\\-3&-6\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&2\\-3&-6\end{matrix}\right))\left(\begin{matrix}4&2\\-3&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\-3&-6\end{matrix}\right))\left(\begin{matrix}18\\27\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\-3&-6\end{matrix}\right))\left(\begin{matrix}18\\27\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\-3&-6\end{matrix}\right))\left(\begin{matrix}18\\27\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{4\left(-6\right)-2\left(-3\right)}&-\frac{2}{4\left(-6\right)-2\left(-3\right)}\\-\frac{-3}{4\left(-6\right)-2\left(-3\right)}&\frac{4}{4\left(-6\right)-2\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}18\\27\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&\frac{1}{9}\\-\frac{1}{6}&-\frac{2}{9}\end{matrix}\right)\left(\begin{matrix}18\\27\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 18+\frac{1}{9}\times 27\\-\frac{1}{6}\times 18-\frac{2}{9}\times 27\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\-9\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=9,y=-9$$

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.

$$4x+2y=18,-3x-6y=27$$

Kia ōrite ai a 4x 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 4.

$$-3\times 4x-3\times 2y=-3\times 18,4\left(-3\right)x+4\left(-6\right)y=4\times 27$$

Whakarūnātia.

$$-12x-6y=-54,-12x-24y=108$$

Me tango -12x-24y=108 mai i -12x-6y=-54 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-12x+12x-6y+24y=-54-108$$

Tāpiri -12x ki te 12x. Ka whakakore atu ngā kupu -12x me 12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-6y+24y=-54-108$$

Tāpiri -6y ki te 24y.

$$18y=-54-108$$

Tāpiri -54 ki te -108.

$$18y=-162$$

Whakawehea ngā taha e rua ki te 18.

$$y=-9$$

Whakaurua te -9 mō y ki -3x-6y=27. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-3x-6\left(-9\right)=27$$

Whakareatia -6 ki te -9.

$$-3x+54=27$$

Me tango 54 mai i ngā taha e rua o te whārite.

$$-3x=-27$$

Whakawehea ngā taha e rua ki te -3.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=-9$$