Whakaoti i te {l}{-2x+15y=-24}{2x+9y=24}

$$\left. \begin{array} { l } { - 2 x + 15 y = - 24 } \\ { 2 x + 9 y = 24 } \end{array} \right.$$

Whakaoti mō x, y

$$x=12$$

$$y=0$$

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.

$$-2x+15y=-24,2x+9y=24$$

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

$$-2x+15y=-24$$

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

$$-2x=-15y-24$$

Whakawehea ngā taha e rua ki te -2.

$$x=-\frac{1}{2}\left(-15y-24\right)$$

Whakareatia -\frac{1}{2} ki te -15y-24.

$$x=\frac{15}{2}y+12$$

Whakakapia te \frac{15y}{2}+12 mō te x ki tērā atu whārite, 2x+9y=24.

$$2\left(\frac{15}{2}y+12\right)+9y=24$$

Whakareatia 2 ki te \frac{15y}{2}+12.

$$15y+24+9y=24$$

Tāpiri 15y ki te 9y.

$$24y+24=24$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$24y=0$$

Whakawehea ngā taha e rua ki te 24.

$$y=0$$

Whakaurua te 0 mō y ki x=\frac{15}{2}y+12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=12$$

Kua oti te pūnaha te whakatau.

$$x=12,y=0$$

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.

$$-2x+15y=-24,2x+9y=24$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-2&15\\2&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-24\\24\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-2&15\\2&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}-2&15\\2&9\end{matrix}\right))\left(\begin{matrix}-2&15\\2&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&15\\2&9\end{matrix}\right))\left(\begin{matrix}-24\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&15\\2&9\end{matrix}\right))\left(\begin{matrix}-24\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&15\\2&9\end{matrix}\right))\left(\begin{matrix}-24\\24\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{-2\times 9-15\times 2}&-\frac{15}{-2\times 9-15\times 2}\\-\frac{2}{-2\times 9-15\times 2}&-\frac{2}{-2\times 9-15\times 2}\end{matrix}\right)\left(\begin{matrix}-24\\24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{16}&\frac{5}{16}\\\frac{1}{24}&\frac{1}{24}\end{matrix}\right)\left(\begin{matrix}-24\\24\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{16}\left(-24\right)+\frac{5}{16}\times 24\\\frac{1}{24}\left(-24\right)+\frac{1}{24}\times 24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=12,y=0$$

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.

$$-2x+15y=-24,2x+9y=24$$

Kia ōrite ai a -2x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -2.

$$2\left(-2\right)x+2\times 15y=2\left(-24\right),-2\times 2x-2\times 9y=-2\times 24$$

Whakarūnātia.

$$-4x+30y=-48,-4x-18y=-48$$

Me tango -4x-18y=-48 mai i -4x+30y=-48 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-4x+4x+30y+18y=-48+48$$

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.

$$30y+18y=-48+48$$

Tāpiri 30y ki te 18y.

$$48y=-48+48$$

Tāpiri -48 ki te 48.

$$48y=0$$

Whakawehea ngā taha e rua ki te 48.

$$y=0$$

Whakaurua te 0 mō y ki 2x+9y=24. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x=24$$

Whakawehea ngā taha e rua ki te 2.

$$x=12$$

Kua oti te pūnaha te whakatau.

$$x=12,y=0$$