Whakaoti i te {l}{6x+7y=-18}{3x-4y=-9}

$$\left. \begin{array} { l } { 6 x + 7 y = - 18 } \\ { 3 x - 4 y = - 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-3$$

$$y=0$$

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.

$$6x+7y=-18,3x-4y=-9$$

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

$$6x+7y=-18$$

Me tango 7y mai i ngā taha e rua o te whārite.

$$6x=-7y-18$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(-7y-18\right)$$

Whakareatia \frac{1}{6} ki te -7y-18.

$$x=-\frac{7}{6}y-3$$

Whakakapia te -\frac{7y}{6}-3 mō te x ki tērā atu whārite, 3x-4y=-9.

$$3\left(-\frac{7}{6}y-3\right)-4y=-9$$

Whakareatia 3 ki te -\frac{7y}{6}-3.

$$-\frac{7}{2}y-9-4y=-9$$

Tāpiri -\frac{7y}{2} ki te -4y.

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

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$-\frac{15}{2}y=0$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{15}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=0$$

Whakaurua te 0 mō y ki x=-\frac{7}{6}y-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=0$$

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.

$$6x+7y=-18,3x-4y=-9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&7\\3&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-18\\-9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&7\\3&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&7\\3&-4\end{matrix}\right))\left(\begin{matrix}6&7\\3&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&7\\3&-4\end{matrix}\right))\left(\begin{matrix}-18\\-9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&7\\3&-4\end{matrix}\right))\left(\begin{matrix}-18\\-9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&7\\3&-4\end{matrix}\right))\left(\begin{matrix}-18\\-9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{6\left(-4\right)-7\times 3}&-\frac{7}{6\left(-4\right)-7\times 3}\\-\frac{3}{6\left(-4\right)-7\times 3}&\frac{6}{6\left(-4\right)-7\times 3}\end{matrix}\right)\left(\begin{matrix}-18\\-9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{45}&\frac{7}{45}\\\frac{1}{15}&-\frac{2}{15}\end{matrix}\right)\left(\begin{matrix}-18\\-9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{45}\left(-18\right)+\frac{7}{45}\left(-9\right)\\\frac{1}{15}\left(-18\right)-\frac{2}{15}\left(-9\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-3,y=0$$

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.

$$6x+7y=-18,3x-4y=-9$$

Kia ōrite ai a 6x 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 6.

$$3\times 6x+3\times 7y=3\left(-18\right),6\times 3x+6\left(-4\right)y=6\left(-9\right)$$

Whakarūnātia.

$$18x+21y=-54,18x-24y=-54$$

Me tango 18x-24y=-54 mai i 18x+21y=-54 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$18x-18x+21y+24y=-54+54$$

Tāpiri 18x ki te -18x. Ka whakakore atu ngā kupu 18x me -18x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$21y+24y=-54+54$$

Tāpiri 21y ki te 24y.

$$45y=-54+54$$

Tāpiri -54 ki te 54.

$$45y=0$$

Whakawehea ngā taha e rua ki te 45.

$$y=0$$

Whakaurua te 0 mō y ki 3x-4y=-9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x=-9$$

Whakawehea ngā taha e rua ki te 3.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=0$$