Whakaoti i te {l}{-2y-3z=3}{-3y-4z=3}

$$\left\{ \begin{array} { l } { - 2 y - 3 z = 3 } \\ { - 3 y - 4 z = 3 } \end{array} \right.$$

Whakaoti mō y, z

$$y=3$$

$$z=-3$$

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.

$$-2y-3z=3,-3y-4z=3$$

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

$$-2y-3z=3$$

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

$$-2y=3z+3$$

Whakawehea ngā taha e rua ki te -2.

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

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

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

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

$$-3\left(-\frac{3}{2}z-\frac{3}{2}\right)-4z=3$$

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

$$\frac{9}{2}z+\frac{9}{2}-4z=3$$

Tāpiri \frac{9z}{2} ki te -4z.

$$\frac{1}{2}z+\frac{9}{2}=3$$

Me tango \frac{9}{2} mai i ngā taha e rua o te whārite.

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

Me whakarea ngā taha e rua ki te 2.

$$z=-3$$

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

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

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

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

Tāpiri -\frac{3}{2} ki te \frac{9}{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.

$$y=3$$

Kua oti te pūnaha te whakatau.

$$y=3,z=-3$$

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.

$$-2y-3z=3,-3y-4z=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-2&-3\\-3&-4\end{matrix}\right)\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}3\\3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-2&-3\\-3&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}-2&-3\\-3&-4\end{matrix}\right))\left(\begin{matrix}-2&-3\\-3&-4\end{matrix}\right)\left(\begin{matrix}y\\z\end{matrix}\right)=inverse(\left(\begin{matrix}-2&-3\\-3&-4\end{matrix}\right))\left(\begin{matrix}3\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\z\end{matrix}\right)=inverse(\left(\begin{matrix}-2&-3\\-3&-4\end{matrix}\right))\left(\begin{matrix}3\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\z\end{matrix}\right)=inverse(\left(\begin{matrix}-2&-3\\-3&-4\end{matrix}\right))\left(\begin{matrix}3\\3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{-2\left(-4\right)-\left(-3\left(-3\right)\right)}&-\frac{-3}{-2\left(-4\right)-\left(-3\left(-3\right)\right)}\\-\frac{-3}{-2\left(-4\right)-\left(-3\left(-3\right)\right)}&-\frac{2}{-2\left(-4\right)-\left(-3\left(-3\right)\right)}\end{matrix}\right)\left(\begin{matrix}3\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}4&-3\\-3&2\end{matrix}\right)\left(\begin{matrix}3\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}4\times 3-3\times 3\\-3\times 3+2\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me z.

$$y=3,z=-3$$

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.

$$-2y-3z=3,-3y-4z=3$$

Kia ōrite ai a -2y me -3y, 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 -2.

$$-3\left(-2\right)y-3\left(-3\right)z=-3\times 3,-2\left(-3\right)y-2\left(-4\right)z=-2\times 3$$

Whakarūnātia.

$$6y+9z=-9,6y+8z=-6$$

Me tango 6y+8z=-6 mai i 6y+9z=-9 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6y-6y+9z-8z=-9+6$$

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

$$9z-8z=-9+6$$

Tāpiri 9z ki te -8z.

$$z=-9+6$$

Tāpiri -9 ki te 6.

$$z=-3$$

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

$$-3y-4\left(-3\right)=3$$

Whakareatia -4 ki te -3.

$$-3y+12=3$$

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

$$-3y=-9$$

Whakawehea ngā taha e rua ki te -3.

$$y=3$$

Kua oti te pūnaha te whakatau.

$$y=3,z=-3$$