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

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

Whakaoti mō x, y

$$x=4$$

$$y=5$$

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.

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

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

$$2x-y=3$$

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

$$2x=y+3$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

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

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

Tāpiri \frac{y}{2} ki te -y.

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

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$-\frac{1}{2}y=-\frac{5}{2}$$

Me whakarea ngā taha e rua ki te -2.

$$y=5$$

Whakaurua te 5 mō y ki x=\frac{1}{2}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.

$$x=\frac{1}{2}\times 5+\frac{3}{2}$$

Whakareatia \frac{1}{2} ki te 5.

$$x=\frac{5+3}{2}$$

Tāpiri \frac{3}{2} ki te \frac{5}{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.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=5$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=5$$

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.

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

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

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

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.

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

Tāpiri 2x ki te -x.

$$x=3+1$$

Tāpiri 3 ki te 1.

$$x=4$$

Whakaurua te 4 mō x ki x-y=-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$4-y=-1$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$-y=-5$$

Kua oti te pūnaha te whakatau.

$$x=4,y=5$$