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

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

Whakaoti mō x, y

$$x = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

$$y=-9$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2.

$$2x-3y=36$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 2.

$$2x-y=18$$

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.

$$2x-3y=36,2x-y=18$$

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

$$2x-3y=36$$

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

$$2x=3y+36$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(3y+36\right)$$

Whakareatia \frac{1}{2} ki te 36+3y.

$$x=\frac{3}{2}y+18$$

Whakakapia te 18+\frac{3y}{2} mō te x ki tērā atu whārite, 2x-y=18.

$$2\left(\frac{3}{2}y+18\right)-y=18$$

Whakareatia 2 ki te 18+\frac{3y}{2}.

$$3y+36-y=18$$

Tāpiri 3y ki te -y.

$$2y+36=18$$

Me tango 36 mai i ngā taha e rua o te whārite.

$$2y=-18$$

Whakawehea ngā taha e rua ki te 2.

$$y=-9$$

Whakaurua te -9 mō y ki x=\frac{3}{2}y+18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{2}\left(-9\right)+18$$

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

$$x=-\frac{27}{2}+18$$

Tāpiri 18 ki te -\frac{27}{2}.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{9}{2},y=-9$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2.

$$2x-3y=36$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 2.

$$2x-y=18$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$2x-3y=36,2x-y=18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-3\\2&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}36\\18\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&-3\\2&-1\end{matrix}\right))\left(\begin{matrix}2&-3\\2&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\2&-1\end{matrix}\right))\left(\begin{matrix}36\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\2&-1\end{matrix}\right))\left(\begin{matrix}36\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\2&-1\end{matrix}\right))\left(\begin{matrix}36\\18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2\left(-1\right)-\left(-3\times 2\right)}&-\frac{-3}{2\left(-1\right)-\left(-3\times 2\right)}\\-\frac{2}{2\left(-1\right)-\left(-3\times 2\right)}&\frac{2}{2\left(-1\right)-\left(-3\times 2\right)}\end{matrix}\right)\left(\begin{matrix}36\\18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&\frac{3}{4}\\-\frac{1}{2}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}36\\18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\times 36+\frac{3}{4}\times 18\\-\frac{1}{2}\times 36+\frac{1}{2}\times 18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{2}\\-9\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{9}{2},y=-9$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2.

$$2x-3y=36$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 2.

$$2x-y=18$$

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.

$$2x-3y=36,2x-y=18$$

Me tango 2x-y=18 mai i 2x-3y=36 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x-3y+y=36-18$$

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

$$-3y+y=36-18$$

Tāpiri -3y ki te y.

$$-2y=36-18$$

Tāpiri 36 ki te -18.

$$-2y=18$$

Whakawehea ngā taha e rua ki te -2.

$$y=-9$$

Whakaurua te -9 mō y ki 2x-y=18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-\left(-9\right)=18$$

Whakareatia -1 ki te -9.

$$2x+9=18$$

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

$$2x=9$$

Whakawehea ngā taha e rua ki te 2.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{9}{2},y=-9$$