Whakaoti i te {c}{-x+5y=1}{-2x-5y=11}

$$\left\{ \begin{array} { c } { - x + 5 y = 1 } \\ { - 2 x - 5 y = 11 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-4$$

$$y=-\frac{3}{5}=-0.6$$

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.

$$-x+5y=1,-2x-5y=11$$

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

$$-x+5y=1$$

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

$$-x=-5y+1$$

Whakawehea ngā taha e rua ki te -1.

$$x=-\left(-5y+1\right)$$

Whakareatia -1 ki te -5y+1.

$$x=5y-1$$

Whakakapia te 5y-1 mō te x ki tērā atu whārite, -2x-5y=11.

$$-2\left(5y-1\right)-5y=11$$

Whakareatia -2 ki te 5y-1.

$$-10y+2-5y=11$$

Tāpiri -10y ki te -5y.

$$-15y+2=11$$

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

$$-15y=9$$

Whakawehea ngā taha e rua ki te -15.

$$y=-\frac{3}{5}$$

Whakaurua te -\frac{3}{5} mō y ki x=5y-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=5\left(-\frac{3}{5}\right)-1$$

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

$$x=-3-1$$

Tāpiri -1 ki te -3.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=-\frac{3}{5}$$

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.

$$-x+5y=1,-2x-5y=11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&5\\-2&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\11\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}-1&5\\-2&-5\end{matrix}\right))\left(\begin{matrix}-1&5\\-2&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&5\\-2&-5\end{matrix}\right))\left(\begin{matrix}1\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&5\\-2&-5\end{matrix}\right))\left(\begin{matrix}1\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&5\\-2&-5\end{matrix}\right))\left(\begin{matrix}1\\11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{-\left(-5\right)-5\left(-2\right)}&-\frac{5}{-\left(-5\right)-5\left(-2\right)}\\-\frac{-2}{-\left(-5\right)-5\left(-2\right)}&-\frac{1}{-\left(-5\right)-5\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}1\\11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&-\frac{1}{3}\\\frac{2}{15}&-\frac{1}{15}\end{matrix}\right)\left(\begin{matrix}1\\11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}-\frac{1}{3}\times 11\\\frac{2}{15}-\frac{1}{15}\times 11\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-4,y=-\frac{3}{5}$$

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.

$$-x+5y=1,-2x-5y=11$$

Kia ōrite ai a -x 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 -1.

$$-2\left(-1\right)x-2\times 5y=-2,-\left(-2\right)x-\left(-5y\right)=-11$$

Whakarūnātia.

$$2x-10y=-2,2x+5y=-11$$

Me tango 2x+5y=-11 mai i 2x-10y=-2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x-10y-5y=-2+11$$

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.

$$-10y-5y=-2+11$$

Tāpiri -10y ki te -5y.

$$-15y=-2+11$$

Tāpiri -2 ki te 11.

$$-15y=9$$

Whakawehea ngā taha e rua ki te -15.

$$y=-\frac{3}{5}$$

Whakaurua te -\frac{3}{5} mō y ki -2x-5y=11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x-5\left(-\frac{3}{5}\right)=11$$

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

$$-2x+3=11$$

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

$$-2x=8$$

Whakawehea ngā taha e rua ki te -2.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=-\frac{3}{5}$$