Whakaoti i te {l}{2x+7y=22}{2x-3y=-14}

$$\left. \begin{array} { l } { 2 x + 7 y = 22 } \\ { 2 x - 3 y = - 14 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{8}{5} = -1\frac{3}{5} = -1.6$$

$$y = \frac{18}{5} = 3\frac{3}{5} = 3.6$$

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.

$$2x+7y=22,2x-3y=-14$$

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

$$2x+7y=22$$

Me tango 7y mai i ngā taha e rua o te whārite.

$$2x=-7y+22$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-7y+22\right)$$

Whakareatia \frac{1}{2} ki te -7y+22.

$$x=-\frac{7}{2}y+11$$

Whakakapia te -\frac{7y}{2}+11 mō te x ki tērā atu whārite, 2x-3y=-14.

$$2\left(-\frac{7}{2}y+11\right)-3y=-14$$

Whakareatia 2 ki te -\frac{7y}{2}+11.

$$-7y+22-3y=-14$$

Tāpiri -7y ki te -3y.

$$-10y+22=-14$$

Me tango 22 mai i ngā taha e rua o te whārite.

$$-10y=-36$$

Whakawehea ngā taha e rua ki te -10.

$$y=\frac{18}{5}$$

Whakaurua te \frac{18}{5} mō y ki x=-\frac{7}{2}y+11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{7}{2}\times \frac{18}{5}+11$$

Whakareatia -\frac{7}{2} ki te \frac{18}{5} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{63}{5}+11$$

Tāpiri 11 ki te -\frac{63}{5}.

$$x=-\frac{8}{5}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{8}{5},y=\frac{18}{5}$$

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.

$$2x+7y=22,2x-3y=-14$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&7\\2&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}22\\-14\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&7\\2&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&7\\2&-3\end{matrix}\right))\left(\begin{matrix}2&7\\2&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&7\\2&-3\end{matrix}\right))\left(\begin{matrix}22\\-14\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&7\\2&-3\end{matrix}\right))\left(\begin{matrix}22\\-14\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&7\\2&-3\end{matrix}\right))\left(\begin{matrix}22\\-14\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{2\left(-3\right)-7\times 2}&-\frac{7}{2\left(-3\right)-7\times 2}\\-\frac{2}{2\left(-3\right)-7\times 2}&\frac{2}{2\left(-3\right)-7\times 2}\end{matrix}\right)\left(\begin{matrix}22\\-14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{20}&\frac{7}{20}\\\frac{1}{10}&-\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}22\\-14\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{20}\times 22+\frac{7}{20}\left(-14\right)\\\frac{1}{10}\times 22-\frac{1}{10}\left(-14\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{5}\\\frac{18}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{8}{5},y=\frac{18}{5}$$

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.

$$2x+7y=22,2x-3y=-14$$

Me tango 2x-3y=-14 mai i 2x+7y=22 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+7y+3y=22+14$$

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.

$$7y+3y=22+14$$

Tāpiri 7y ki te 3y.

$$10y=22+14$$

Tāpiri 22 ki te 14.

$$10y=36$$

Whakawehea ngā taha e rua ki te 10.

$$y=\frac{18}{5}$$

Whakaurua te \frac{18}{5} mō y ki 2x-3y=-14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-3\times \frac{18}{5}=-14$$

Whakareatia -3 ki te \frac{18}{5}.

$$2x-\frac{54}{5}=-14$$

Me tāpiri \frac{54}{5} ki ngā taha e rua o te whārite.

$$2x=-\frac{16}{5}$$

Whakawehea ngā taha e rua ki te 2.

$$x=-\frac{8}{5}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{8}{5},y=\frac{18}{5}$$