Whakaoti i te {l}{2x+y=-1}{3x+y=0}

$$\left\{ \begin{array} { l } { 2 x + y = - 1 } \\ { 3 x + y = 0 } \end{array} \right.$$

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.

$$2x+y=-1,3x+y=0$$

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

$$2x+y=-1$$

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

$$2x=-y-1$$

Whakawehea ngā taha e rua ki te 2.

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

Whakareatia \frac{1}{2} ki te -y-1.

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

Whakakapia te \frac{-y-1}{2} mō te x ki tērā atu whārite, 3x+y=0.

$$3\left(-\frac{1}{2}y-\frac{1}{2}\right)+y=0$$

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

$$-\frac{3}{2}y-\frac{3}{2}+y=0$$

Tāpiri -\frac{3y}{2} ki te y.

$$-\frac{1}{2}y-\frac{3}{2}=0$$

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

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

Me whakarea ngā taha e rua ki te -2.

$$y=-3$$

Whakaurua te -3 mō y ki x=-\frac{1}{2}y-\frac{1}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

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

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

Tāpiri -\frac{1}{2} ki te \frac{3}{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.

$$2x+y=-1,3x+y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

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.

$$2x+y=-1,3x+y=0$$

Me tango 3x+y=0 mai i 2x+y=-1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-3x+y-y=-1$$

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

$$2x-3x=-1$$

Tāpiri 2x ki te -3x.

$$-x=-1$$

Whakawehea ngā taha e rua ki te -1.

$$x=1$$

Whakaurua te 1 mō x ki 3x+y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$3+y=0$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$y=-3$$

Kua oti te pūnaha te whakatau.

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