Whakaoti i te {l}{x+2y=-12}{3x-y=-1}

$$\left\{ \begin{array} { l } { x + 2 y = - 12 } \\ { 3 x - y = - 1 } \end{array} \right.$$

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.

$$x+2y=-12,3x-y=-1$$

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

$$x+2y=-12$$

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

$$x=-2y-12$$

Whakakapia te -2y-12 mō te x ki tērā atu whārite, 3x-y=-1.

$$3\left(-2y-12\right)-y=-1$$

Whakareatia 3 ki te -2y-12.

$$-6y-36-y=-1$$

Tāpiri -6y ki te -y.

$$-7y-36=-1$$

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

$$-7y=35$$

Whakawehea ngā taha e rua ki te -7.

$$y=-5$$

Whakaurua te -5 mō y ki x=-2y-12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\left(-5\right)-12$$

Whakareatia -2 ki te -5.

$$x=10-12$$

Tāpiri -12 ki te 10.

$$x=-2$$

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.

$$x+2y=-12,3x-y=-1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\3&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-12\\-1\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&2\\3&-1\end{matrix}\right))\left(\begin{matrix}1&2\\3&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\3&-1\end{matrix}\right))\left(\begin{matrix}-12\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\3&-1\end{matrix}\right))\left(\begin{matrix}-12\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\3&-1\end{matrix}\right))\left(\begin{matrix}-12\\-1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-2\times 3}&-\frac{2}{-1-2\times 3}\\-\frac{3}{-1-2\times 3}&\frac{1}{-1-2\times 3}\end{matrix}\right)\left(\begin{matrix}-12\\-1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}&\frac{2}{7}\\\frac{3}{7}&-\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}-12\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}\left(-12\right)+\frac{2}{7}\left(-1\right)\\\frac{3}{7}\left(-12\right)-\frac{1}{7}\left(-1\right)\end{matrix}\right)$$

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.

$$x+2y=-12,3x-y=-1$$

Kia ōrite ai a x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$3x+3\times 2y=3\left(-12\right),3x-y=-1$$

Whakarūnātia.

$$3x+6y=-36,3x-y=-1$$

Me tango 3x-y=-1 mai i 3x+6y=-36 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+6y+y=-36+1$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y+y=-36+1$$

Tāpiri 6y ki te y.

$$7y=-36+1$$

Tāpiri -36 ki te 1.

$$7y=-35$$

Whakawehea ngā taha e rua ki te 7.

$$y=-5$$

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

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

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

$$3x=-6$$

Whakawehea ngā taha e rua ki te 3.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=-5$$