Whakaoti i te {l}{x+2y=3(1+y)+1}{8-y=2(1-y)+3x}

$$\left. \begin{array}{l}{ x + 2 y = 3 ( 1 + y ) + 1 }\\{ 8 - y = 2 ( 1 - y ) + 3 x }\end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=-3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 1+y.

$$x+2y=3+3y+1$$

Tāpirihia te 3 ki te 1, ka 4.

$$x+2y=4+3y$$

Tangohia te 3y mai i ngā taha e rua.

$$x+2y-3y=4$$

Pahekotia te 2y me -3y, ka -y.

$$x-y=4$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 1-y.

$$8-y=2-2y+3x$$

Me tāpiri te 2y ki ngā taha e rua.

$$8-y+2y=2+3x$$

Pahekotia te -y me 2y, ka y.

$$8+y=2+3x$$

Tangohia te 3x mai i ngā taha e rua.

$$8+y-3x=2$$

Tangohia te 8 mai i ngā taha e rua.

$$y-3x=2-8$$

Tangohia te 8 i te 2, ka -6.

$$y-3x=-6$$

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.

$$x-y=4,-3x+y=-6$$

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

$$x-y=4$$

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

$$x=y+4$$

Whakakapia te y+4 mō te x ki tērā atu whārite, -3x+y=-6.

$$-3\left(y+4\right)+y=-6$$

Whakareatia -3 ki te y+4.

$$-3y-12+y=-6$$

Tāpiri -3y ki te y.

$$-2y-12=-6$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$-2y=6$$

Whakawehea ngā taha e rua ki te -2.

$$y=-3$$

Whakaurua te -3 mō y ki x=y+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-3+4$$

Tāpiri 4 ki te -3.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-3$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 1+y.

$$x+2y=3+3y+1$$

Tāpirihia te 3 ki te 1, ka 4.

$$x+2y=4+3y$$

Tangohia te 3y mai i ngā taha e rua.

$$x+2y-3y=4$$

Pahekotia te 2y me -3y, ka -y.

$$x-y=4$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 1-y.

$$8-y=2-2y+3x$$

Me tāpiri te 2y ki ngā taha e rua.

$$8-y+2y=2+3x$$

Pahekotia te -y me 2y, ka y.

$$8+y=2+3x$$

Tangohia te 3x mai i ngā taha e rua.

$$8+y-3x=2$$

Tangohia te 8 mai i ngā taha e rua.

$$y-3x=2-8$$

Tangohia te 8 i te 2, ka -6.

$$y-3x=-6$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$x-y=4,-3x+y=-6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\-3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\-6\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-1\\-3&1\end{matrix}\right))\left(\begin{matrix}1&-1\\-3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\-3&1\end{matrix}\right))\left(\begin{matrix}4\\-6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\-3&1\end{matrix}\right))\left(\begin{matrix}4\\-6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\-3&1\end{matrix}\right))\left(\begin{matrix}4\\-6\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-\left(-\left(-3\right)\right)}&-\frac{-1}{1-\left(-\left(-3\right)\right)}\\-\frac{-3}{1-\left(-\left(-3\right)\right)}&\frac{1}{1-\left(-\left(-3\right)\right)}\end{matrix}\right)\left(\begin{matrix}4\\-6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}&-\frac{1}{2}\\-\frac{3}{2}&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}4\\-6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}\times 4-\frac{1}{2}\left(-6\right)\\-\frac{3}{2}\times 4-\frac{1}{2}\left(-6\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=-3$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 1+y.

$$x+2y=3+3y+1$$

Tāpirihia te 3 ki te 1, ka 4.

$$x+2y=4+3y$$

Tangohia te 3y mai i ngā taha e rua.

$$x+2y-3y=4$$

Pahekotia te 2y me -3y, ka -y.

$$x-y=4$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 1-y.

$$8-y=2-2y+3x$$

Me tāpiri te 2y ki ngā taha e rua.

$$8-y+2y=2+3x$$

Pahekotia te -y me 2y, ka y.

$$8+y=2+3x$$

Tangohia te 3x mai i ngā taha e rua.

$$8+y-3x=2$$

Tangohia te 8 mai i ngā taha e rua.

$$y-3x=2-8$$

Tangohia te 8 i te 2, ka -6.

$$y-3x=-6$$

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.

$$x-y=4,-3x+y=-6$$

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.

$$-3x-3\left(-1\right)y=-3\times 4,-3x+y=-6$$

Whakarūnātia.

$$-3x+3y=-12,-3x+y=-6$$

Me tango -3x+y=-6 mai i -3x+3y=-12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-3x+3x+3y-y=-12+6$$

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.

$$3y-y=-12+6$$

Tāpiri 3y ki te -y.

$$2y=-12+6$$

Tāpiri -12 ki te 6.

$$2y=-6$$

Whakawehea ngā taha e rua ki te 2.

$$y=-3$$

Whakaurua te -3 mō y ki -3x+y=-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-3x-3=-6$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$-3x=-3$$

Whakawehea ngā taha e rua ki te -3.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-3$$