Whakaoti i te {l}{6x+3y=9}{x-4y=15}

$$\left. \begin{array} { l } { 6 x + 3 y = 9 } \\ { x - 4 y = 15 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=-3$$

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.

$$6x+3y=9,x-4y=15$$

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

$$6x+3y=9$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$6x=-3y+9$$

Whakawehea ngā taha e rua ki te 6.

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

Whakareatia \frac{1}{6} ki te -3y+9.

$$x=-\frac{1}{2}y+\frac{3}{2}$$

Whakakapia te \frac{-y+3}{2} mō te x ki tērā atu whārite, x-4y=15.

$$-\frac{1}{2}y+\frac{3}{2}-4y=15$$

Tāpiri -\frac{y}{2} ki te -4y.

$$-\frac{9}{2}y+\frac{3}{2}=15$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

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

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.

$$y=-3$$

Whakaurua te -3 mō y ki x=-\frac{1}{2}y+\frac{3}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\left(-3\right)+\frac{3}{2}$$

Whakareatia -\frac{1}{2} ki te -3.

$$x=\frac{3+3}{2}$$

Tāpiri \frac{3}{2} ki te \frac{3}{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.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=-3$$

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.

$$6x+3y=9,x-4y=15$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&3\\1&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\15\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&3\\1&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&3\\1&-4\end{matrix}\right))\left(\begin{matrix}6&3\\1&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&3\\1&-4\end{matrix}\right))\left(\begin{matrix}9\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&3\\1&-4\end{matrix}\right))\left(\begin{matrix}9\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&3\\1&-4\end{matrix}\right))\left(\begin{matrix}9\\15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{6\left(-4\right)-3}&-\frac{3}{6\left(-4\right)-3}\\-\frac{1}{6\left(-4\right)-3}&\frac{6}{6\left(-4\right)-3}\end{matrix}\right)\left(\begin{matrix}9\\15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{27}&\frac{1}{9}\\\frac{1}{27}&-\frac{2}{9}\end{matrix}\right)\left(\begin{matrix}9\\15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{27}\times 9+\frac{1}{9}\times 15\\\frac{1}{27}\times 9-\frac{2}{9}\times 15\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=-3$$

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.

$$6x+3y=9,x-4y=15$$

Kia ōrite ai a 6x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 6.

$$6x+3y=9,6x+6\left(-4\right)y=6\times 15$$

Whakarūnātia.

$$6x+3y=9,6x-24y=90$$

Me tango 6x-24y=90 mai i 6x+3y=9 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+3y+24y=9-90$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3y+24y=9-90$$

Tāpiri 3y ki te 24y.

$$27y=9-90$$

Tāpiri 9 ki te -90.

$$27y=-81$$

Whakawehea ngā taha e rua ki te 27.

$$y=-3$$

Whakaurua te -3 mō y ki x-4y=15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-4\left(-3\right)=15$$

Whakareatia -4 ki te -3.

$$x+12=15$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=-3$$