Whakaoti i te {l}{x/6-y=-1}{3x-2y=6}

$$\left\{ \begin{array} { l } { \frac { x } { 6 } - y = - 1 } \\ { 3 x - 2 y = 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

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.

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

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

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

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

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

Me whakarea ngā taha e rua ki te 6.

$$x=6\left(y-1\right)$$

Whakareatia 6 ki te y-1.

$$x=6y-6$$

Whakakapia te -6+6y mō te x ki tērā atu whārite, 3x-2y=6.

$$3\left(6y-6\right)-2y=6$$

Whakareatia 3 ki te -6+6y.

$$18y-18-2y=6$$

Tāpiri 18y ki te -2y.

$$16y-18=6$$

Me tāpiri 18 ki ngā taha e rua o te whārite.

$$16y=24$$

Whakawehea ngā taha e rua ki te 16.

$$y=\frac{3}{2}$$

Whakaurua te \frac{3}{2} mō y ki x=6y-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=6\times \frac{3}{2}-6$$

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

$$x=9-6$$

Tāpiri -6 ki te 9.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=\frac{3}{2}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{4}&\frac{3}{8}\\-\frac{9}{8}&\frac{1}{16}\end{matrix}\right)\left(\begin{matrix}-1\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{4}\left(-1\right)+\frac{3}{8}\times 6\\-\frac{9}{8}\left(-1\right)+\frac{1}{16}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\\frac{3}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=\frac{3}{2}$$

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.

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

Kia ōrite ai a \frac{x}{6} 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 \frac{1}{6}.

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

Whakarūnātia.

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

Me tango \frac{1}{2}x-\frac{1}{3}y=1 mai i \frac{1}{2}x-3y=-3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

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

Tāpiri \frac{x}{2} ki te -\frac{x}{2}. Ka whakakore atu ngā kupu \frac{x}{2} me -\frac{x}{2}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

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

Tāpiri -3y ki te \frac{y}{3}.

$$-\frac{8}{3}y=-3-1$$

Tāpiri -3 ki te -1.

$$-\frac{8}{3}y=-4$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{8}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{3}{2}$$

Whakaurua te \frac{3}{2} mō y ki 3x-2y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-2\times \frac{3}{2}=6$$

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

$$3x-3=6$$

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

$$3x=9$$

Whakawehea ngā taha e rua ki te 3.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=\frac{3}{2}$$