Whakaoti i te {l}{x+2y+3=0}{4x+5y+6=0}

$$\left. \begin{array} { l } { x + 2 y + 3 = 0 } \\ { 4 x + 5 y + 6 = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=-2$$

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.

$$x+2y+3=0,4x+5y+6=0$$

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

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

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

$$x+2y=-3$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$x=-2y-3$$

Whakakapia te -2y-3 mō te x ki tērā atu whārite, 4x+5y+6=0.

$$4\left(-2y-3\right)+5y+6=0$$

Whakareatia 4 ki te -2y-3.

$$-8y-12+5y+6=0$$

Tāpiri -8y ki te 5y.

$$-3y-12+6=0$$

Tāpiri -12 ki te 6.

$$-3y-6=0$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$-3y=6$$

Whakawehea ngā taha e rua ki te -3.

$$y=-2$$

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

$$x=-2\left(-2\right)-3$$

Whakareatia -2 ki te -2.

$$x=4-3$$

Tāpiri -3 ki te 4.

$$x=1$$

Kua oti te pūnaha te whakatau.

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

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.

$$x+2y+3=0,4x+5y+6=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\4&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\-6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&2\\4&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&2\\4&5\end{matrix}\right))\left(\begin{matrix}1&2\\4&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\4&5\end{matrix}\right))\left(\begin{matrix}-3\\-6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\4&5\end{matrix}\right))\left(\begin{matrix}-3\\-6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\4&5\end{matrix}\right))\left(\begin{matrix}-3\\-6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5-2\times 4}&-\frac{2}{5-2\times 4}\\-\frac{4}{5-2\times 4}&\frac{1}{5-2\times 4}\end{matrix}\right)\left(\begin{matrix}-3\\-6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{3}&\frac{2}{3}\\\frac{4}{3}&-\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}-3\\-6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{3}\left(-3\right)+\frac{2}{3}\left(-6\right)\\\frac{4}{3}\left(-3\right)-\frac{1}{3}\left(-6\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$x+2y+3=0,4x+5y+6=0$$

Kia ōrite ai a x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$4x+4\times 2y+4\times 3=0,4x+5y+6=0$$

Whakarūnātia.

$$4x+8y+12=0,4x+5y+6=0$$

Me tango 4x+5y+6=0 mai i 4x+8y+12=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-4x+8y-5y+12-6=0$$

Tāpiri 4x ki te -4x. Ka whakakore atu ngā kupu 4x me -4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$8y-5y+12-6=0$$

Tāpiri 8y ki te -5y.

$$3y+12-6=0$$

Tāpiri 12 ki te -6.

$$3y+6=0$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$3y=-6$$

Whakawehea ngā taha e rua ki te 3.

$$y=-2$$

Whakaurua te -2 mō y ki 4x+5y+6=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+5\left(-2\right)+6=0$$

Whakareatia 5 ki te -2.

$$4x-10+6=0$$

Tāpiri -10 ki te 6.

$$4x-4=0$$

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

$$4x=4$$

Whakawehea ngā taha e rua ki te 4.

$$x=1$$

Kua oti te pūnaha te whakatau.

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