Whakaoti i te {c}{5x-y=5}{3y-2x=11}

$$\left\{ \begin{array} { c } { 5 x - y = 5 } \\ { 3 y - 2 x = 11 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=5$$

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.

$$5x-y=5,-2x+3y=11$$

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

$$5x-y=5$$

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

$$5x=y+5$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(y+5\right)$$

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

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

Whakakapia te \frac{y}{5}+1 mō te x ki tērā atu whārite, -2x+3y=11.

$$-2\left(\frac{1}{5}y+1\right)+3y=11$$

Whakareatia -2 ki te \frac{y}{5}+1.

$$-\frac{2}{5}y-2+3y=11$$

Tāpiri -\frac{2y}{5} ki te 3y.

$$\frac{13}{5}y-2=11$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$\frac{13}{5}y=13$$

Whakawehea ngā taha e rua o te whārite ki te \frac{13}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=5$$

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

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

Whakareatia \frac{1}{5} ki te 5.

$$x=1+1$$

Tāpiri 1 ki te 1.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=5$$

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.

$$5x-y=5,-2x+3y=11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-1\\-2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\11\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-1\\-2&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-1\\-2&3\end{matrix}\right))\left(\begin{matrix}5&-1\\-2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\-2&3\end{matrix}\right))\left(\begin{matrix}5\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\-2&3\end{matrix}\right))\left(\begin{matrix}5\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\-2&3\end{matrix}\right))\left(\begin{matrix}5\\11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5\times 3-\left(-\left(-2\right)\right)}&-\frac{-1}{5\times 3-\left(-\left(-2\right)\right)}\\-\frac{-2}{5\times 3-\left(-\left(-2\right)\right)}&\frac{5}{5\times 3-\left(-\left(-2\right)\right)}\end{matrix}\right)\left(\begin{matrix}5\\11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{13}&\frac{1}{13}\\\frac{2}{13}&\frac{5}{13}\end{matrix}\right)\left(\begin{matrix}5\\11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{13}\times 5+\frac{1}{13}\times 11\\\frac{2}{13}\times 5+\frac{5}{13}\times 11\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=2,y=5$$

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.

$$5x-y=5,-2x+3y=11$$

Kia ōrite ai a 5x 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 5.

$$-2\times 5x-2\left(-1\right)y=-2\times 5,5\left(-2\right)x+5\times 3y=5\times 11$$

Whakarūnātia.

$$-10x+2y=-10,-10x+15y=55$$

Me tango -10x+15y=55 mai i -10x+2y=-10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-10x+10x+2y-15y=-10-55$$

Tāpiri -10x ki te 10x. Ka whakakore atu ngā kupu -10x me 10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2y-15y=-10-55$$

Tāpiri 2y ki te -15y.

$$-13y=-10-55$$

Tāpiri -10 ki te -55.

$$-13y=-65$$

Whakawehea ngā taha e rua ki te -13.

$$y=5$$

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

$$-2x+3\times 5=11$$

Whakareatia 3 ki te 5.

$$-2x+15=11$$

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

$$-2x=-4$$

Whakawehea ngā taha e rua ki te -2.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=5$$