Whakaoti i te {l}{2x+y-3=0}{2x-3y-7=0}

$$\left. \begin{array} { l } { 2 x + y - 3 = 0 } \\ { 2 x - 3 y - 7 = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=-1$$

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.

$$2x+y-3=0,2x-3y-7=0$$

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

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

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

$$2x+y=3$$

Me tango y mai i 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{-y+3}{2} mō te x ki tērā atu whārite, 2x-3y-7=0.

$$2\left(-\frac{1}{2}y+\frac{3}{2}\right)-3y-7=0$$

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

$$-y+3-3y-7=0$$

Tāpiri -y ki te -3y.

$$-4y+3-7=0$$

Tāpiri 3 ki te -7.

$$-4y-4=0$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$-4y=4$$

Whakawehea ngā taha e rua ki te -4.

$$y=-1$$

Whakaurua te -1 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.

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

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

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

Tāpiri \frac{3}{2} ki te \frac{1}{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.

$$x=2$$

Kua oti te pūnaha te whakatau.

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

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.

$$2x+y-3=0,2x-3y-7=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&1\\2&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\7\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&1\\2&-3\end{matrix}\right))\left(\begin{matrix}2&1\\2&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\2&-3\end{matrix}\right))\left(\begin{matrix}3\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\2&-3\end{matrix}\right))\left(\begin{matrix}3\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\2&-3\end{matrix}\right))\left(\begin{matrix}3\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{2\left(-3\right)-2}&-\frac{1}{2\left(-3\right)-2}\\-\frac{2}{2\left(-3\right)-2}&\frac{2}{2\left(-3\right)-2}\end{matrix}\right)\left(\begin{matrix}3\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8}&\frac{1}{8}\\\frac{1}{4}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}3\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8}\times 3+\frac{1}{8}\times 7\\\frac{1}{4}\times 3-\frac{1}{4}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$2x+y-3=0,2x-3y-7=0$$

Me tango 2x-3y-7=0 mai i 2x+y-3=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+y+3y-3+7=0$$

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.

$$y+3y-3+7=0$$

Tāpiri y ki te 3y.

$$4y-3+7=0$$

Tāpiri -3 ki te 7.

$$4y+4=0$$

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

$$4y=-4$$

Whakawehea ngā taha e rua ki te 4.

$$y=-1$$

Whakaurua te -1 mō y ki 2x-3y-7=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-3\left(-1\right)-7=0$$

Whakareatia -3 ki te -1.

$$2x+3-7=0$$

Tāpiri 3 ki te -7.

$$2x-4=0$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$2x=4$$

Whakawehea ngā taha e rua ki te 2.

$$x=2$$

Kua oti te pūnaha te whakatau.

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