Whakaoti i te {l}{-x+5y=15}{4x+10y=-2}

$$\left\{ \begin{array} { l } { - x + 5 y = 15 } \\ { 4 x + 10 y = - 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{16}{3} = -5\frac{1}{3} \approx -5.333333333$$

$$y = \frac{29}{15} = 1\frac{14}{15} \approx 1.933333333$$

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.

$$-x+5y=15,4x+10y=-2$$

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

$$-x+5y=15$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$-x=-5y+15$$

Whakawehea ngā taha e rua ki te -1.

$$x=-\left(-5y+15\right)$$

Whakareatia -1 ki te -5y+15.

$$x=5y-15$$

Whakakapia te -15+5y mō te x ki tērā atu whārite, 4x+10y=-2.

$$4\left(5y-15\right)+10y=-2$$

Whakareatia 4 ki te -15+5y.

$$20y-60+10y=-2$$

Tāpiri 20y ki te 10y.

$$30y-60=-2$$

Me tāpiri 60 ki ngā taha e rua o te whārite.

$$30y=58$$

Whakawehea ngā taha e rua ki te 30.

$$y=\frac{29}{15}$$

Whakaurua te \frac{29}{15} mō y ki x=5y-15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=5\times \frac{29}{15}-15$$

Whakareatia 5 ki te \frac{29}{15}.

$$x=\frac{29}{3}-15$$

Tāpiri -15 ki te \frac{29}{3}.

$$x=-\frac{16}{3}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{16}{3},y=\frac{29}{15}$$

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.

$$-x+5y=15,4x+10y=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&5\\4&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\-2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-1&5\\4&10\end{matrix}\right).

$$inverse(\left(\begin{matrix}-1&5\\4&10\end{matrix}\right))\left(\begin{matrix}-1&5\\4&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&5\\4&10\end{matrix}\right))\left(\begin{matrix}15\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&5\\4&10\end{matrix}\right))\left(\begin{matrix}15\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&5\\4&10\end{matrix}\right))\left(\begin{matrix}15\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{-10-5\times 4}&-\frac{5}{-10-5\times 4}\\-\frac{4}{-10-5\times 4}&-\frac{1}{-10-5\times 4}\end{matrix}\right)\left(\begin{matrix}15\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&\frac{1}{6}\\\frac{2}{15}&\frac{1}{30}\end{matrix}\right)\left(\begin{matrix}15\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}\times 15+\frac{1}{6}\left(-2\right)\\\frac{2}{15}\times 15+\frac{1}{30}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{16}{3}\\\frac{29}{15}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{16}{3},y=\frac{29}{15}$$

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.

$$-x+5y=15,4x+10y=-2$$

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.

$$4\left(-1\right)x+4\times 5y=4\times 15,-4x-10y=-\left(-2\right)$$

Whakarūnātia.

$$-4x+20y=60,-4x-10y=2$$

Me tango -4x-10y=2 mai i -4x+20y=60 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-4x+4x+20y+10y=60-2$$

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.

$$20y+10y=60-2$$

Tāpiri 20y ki te 10y.

$$30y=60-2$$

Tāpiri 60 ki te -2.

$$30y=58$$

Whakawehea ngā taha e rua ki te 30.

$$y=\frac{29}{15}$$

Whakaurua te \frac{29}{15} mō y ki 4x+10y=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+10\times \frac{29}{15}=-2$$

Whakareatia 10 ki te \frac{29}{15}.

$$4x+\frac{58}{3}=-2$$

Me tango \frac{58}{3} mai i ngā taha e rua o te whārite.

$$4x=-\frac{64}{3}$$

Whakawehea ngā taha e rua ki te 4.

$$x=-\frac{16}{3}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{16}{3},y=\frac{29}{15}$$