Whakaoti i te {l}{-10x+3y=-2}{4x-y=8}

$$\left\{ \begin{array} { l } { - 10 x + 3 y = - 2 } \\ { 4 x - y = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=11$$

$$y=36$$

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.

$$-10x+3y=-2,4x-y=8$$

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

$$-10x+3y=-2$$

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

$$-10x=-3y-2$$

Whakawehea ngā taha e rua ki te -10.

$$x=-\frac{1}{10}\left(-3y-2\right)$$

Whakareatia -\frac{1}{10} ki te -3y-2.

$$x=\frac{3}{10}y+\frac{1}{5}$$

Whakakapia te \frac{3y}{10}+\frac{1}{5} mō te x ki tērā atu whārite, 4x-y=8.

$$4\left(\frac{3}{10}y+\frac{1}{5}\right)-y=8$$

Whakareatia 4 ki te \frac{3y}{10}+\frac{1}{5}.

$$\frac{6}{5}y+\frac{4}{5}-y=8$$

Tāpiri \frac{6y}{5} ki te -y.

$$\frac{1}{5}y+\frac{4}{5}=8$$

Me tango \frac{4}{5} mai i ngā taha e rua o te whārite.

$$\frac{1}{5}y=\frac{36}{5}$$

Me whakarea ngā taha e rua ki te 5.

$$y=36$$

Whakaurua te 36 mō y ki x=\frac{3}{10}y+\frac{1}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{10}\times 36+\frac{1}{5}$$

Whakareatia \frac{3}{10} ki te 36.

$$x=\frac{54+1}{5}$$

Tāpiri \frac{1}{5} ki te \frac{54}{5} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=11$$

Kua oti te pūnaha te whakatau.

$$x=11,y=36$$

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.

$$-10x+3y=-2,4x-y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-10&3\\4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2\\8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-10&3\\4&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}-10&3\\4&-1\end{matrix}\right))\left(\begin{matrix}-10&3\\4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-10&3\\4&-1\end{matrix}\right))\left(\begin{matrix}-2\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-10&3\\4&-1\end{matrix}\right))\left(\begin{matrix}-2\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-10&3\\4&-1\end{matrix}\right))\left(\begin{matrix}-2\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-10\left(-1\right)-3\times 4}&-\frac{3}{-10\left(-1\right)-3\times 4}\\-\frac{4}{-10\left(-1\right)-3\times 4}&-\frac{10}{-10\left(-1\right)-3\times 4}\end{matrix}\right)\left(\begin{matrix}-2\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{3}{2}\\2&5\end{matrix}\right)\left(\begin{matrix}-2\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\left(-2\right)+\frac{3}{2}\times 8\\2\left(-2\right)+5\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11\\36\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=11,y=36$$

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.

$$-10x+3y=-2,4x-y=8$$

Kia ōrite ai a -10x 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 -10.

$$4\left(-10\right)x+4\times 3y=4\left(-2\right),-10\times 4x-10\left(-1\right)y=-10\times 8$$

Whakarūnātia.

$$-40x+12y=-8,-40x+10y=-80$$

Me tango -40x+10y=-80 mai i -40x+12y=-8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-40x+40x+12y-10y=-8+80$$

Tāpiri -40x ki te 40x. Ka whakakore atu ngā kupu -40x me 40x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$12y-10y=-8+80$$

Tāpiri 12y ki te -10y.

$$2y=-8+80$$

Tāpiri -8 ki te 80.

$$2y=72$$

Whakawehea ngā taha e rua ki te 2.

$$y=36$$

Whakaurua te 36 mō y ki 4x-y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-36=8$$

Me tāpiri 36 ki ngā taha e rua o te whārite.

$$4x=44$$

Whakawehea ngā taha e rua ki te 4.

$$x=11$$

Kua oti te pūnaha te whakatau.

$$x=11,y=36$$