Whakaoti i te {l}{x+2y=15}{x-2y=-7}

$$\left. \begin{array} { l } { x + 2 y = 15 } \\ { x - 2 y = - 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y = \frac{11}{2} = 5\frac{1}{2} = 5.5$$

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.

$$x+2y=15,x-2y=-7$$

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

$$x+2y=15$$

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

$$x=-2y+15$$

Whakakapia te -2y+15 mō te x ki tērā atu whārite, x-2y=-7.

$$-2y+15-2y=-7$$

Tāpiri -2y ki te -2y.

$$-4y+15=-7$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$-4y=-22$$

Whakawehea ngā taha e rua ki te -4.

$$y=\frac{11}{2}$$

Whakaurua te \frac{11}{2} mō y ki x=-2y+15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times \frac{11}{2}+15$$

Whakareatia -2 ki te \frac{11}{2}.

$$x=-11+15$$

Tāpiri 15 ki te -11.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=\frac{11}{2}$$

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.

$$x+2y=15,x-2y=-7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\-7\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&2\\1&-2\end{matrix}\right))\left(\begin{matrix}1&2\\1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\1&-2\end{matrix}\right))\left(\begin{matrix}15\\-7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\1&-2\end{matrix}\right))\left(\begin{matrix}15\\-7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\1&-2\end{matrix}\right))\left(\begin{matrix}15\\-7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-2}&-\frac{2}{-2-2}\\-\frac{1}{-2-2}&\frac{1}{-2-2}\end{matrix}\right)\left(\begin{matrix}15\\-7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{2}\\\frac{1}{4}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}15\\-7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 15+\frac{1}{2}\left(-7\right)\\\frac{1}{4}\times 15-\frac{1}{4}\left(-7\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\\frac{11}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=\frac{11}{2}$$

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.

$$x+2y=15,x-2y=-7$$

Me tango x-2y=-7 mai i x+2y=15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+2y+2y=15+7$$

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

$$2y+2y=15+7$$

Tāpiri 2y ki te 2y.

$$4y=15+7$$

Tāpiri 15 ki te 7.

$$4y=22$$

Whakawehea ngā taha e rua ki te 4.

$$y=\frac{11}{2}$$

Whakaurua te \frac{11}{2} mō y ki x-2y=-7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-2\times \frac{11}{2}=-7$$

Whakareatia -2 ki te \frac{11}{2}.

$$x-11=-7$$

Me tāpiri 11 ki ngā taha e rua o te whārite.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=\frac{11}{2}$$