Whakaoti i te {l}{6x+y=-9}{2x-3y=7}

$$\left\{ \begin{array} { l } { 6 x + y = - 9 } \\ { 2 x - 3 y = 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-1$$

$$y=-3$$

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.

$$6x+y=-9,2x-3y=7$$

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

$$6x+y=-9$$

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

$$6x=-y-9$$

Whakawehea ngā taha e rua ki te 6.

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

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

$$x=-\frac{1}{6}y-\frac{3}{2}$$

Whakakapia te -\frac{y}{6}-\frac{3}{2} mō te x ki tērā atu whārite, 2x-3y=7.

$$2\left(-\frac{1}{6}y-\frac{3}{2}\right)-3y=7$$

Whakareatia 2 ki te -\frac{y}{6}-\frac{3}{2}.

$$-\frac{1}{3}y-3-3y=7$$

Tāpiri -\frac{y}{3} ki te -3y.

$$-\frac{10}{3}y-3=7$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$-\frac{10}{3}y=10$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{10}{3}, 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}{6}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.

$$x=-\frac{1}{6}\left(-3\right)-\frac{3}{2}$$

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

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

Tāpiri -\frac{3}{2} ki te \frac{1}{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.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=-3$$

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.

$$6x+y=-9,2x-3y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&1\\2&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-9\\7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&1\\2&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&1\\2&-3\end{matrix}\right))\left(\begin{matrix}6&1\\2&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&1\\2&-3\end{matrix}\right))\left(\begin{matrix}-9\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&1\\2&-3\end{matrix}\right))\left(\begin{matrix}-9\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&1\\2&-3\end{matrix}\right))\left(\begin{matrix}-9\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{6\left(-3\right)-2}&-\frac{1}{6\left(-3\right)-2}\\-\frac{2}{6\left(-3\right)-2}&\frac{6}{6\left(-3\right)-2}\end{matrix}\right)\left(\begin{matrix}-9\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{20}&\frac{1}{20}\\\frac{1}{10}&-\frac{3}{10}\end{matrix}\right)\left(\begin{matrix}-9\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{20}\left(-9\right)+\frac{1}{20}\times 7\\\frac{1}{10}\left(-9\right)-\frac{3}{10}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\-3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=-3$$

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.

$$6x+y=-9,2x-3y=7$$

Kia ōrite ai a 6x 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 6.

$$2\times 6x+2y=2\left(-9\right),6\times 2x+6\left(-3\right)y=6\times 7$$

Whakarūnātia.

$$12x+2y=-18,12x-18y=42$$

Me tango 12x-18y=42 mai i 12x+2y=-18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+2y+18y=-18-42$$

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.

$$2y+18y=-18-42$$

Tāpiri 2y ki te 18y.

$$20y=-18-42$$

Tāpiri -18 ki te -42.

$$20y=-60$$

Whakawehea ngā taha e rua ki te 20.

$$y=-3$$

Whakaurua te -3 mō y ki 2x-3y=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-3\left(-3\right)=7$$

Whakareatia -3 ki te -3.

$$2x+9=7$$

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

$$2x=-2$$

Whakawehea ngā taha e rua ki te 2.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=-3$$