Whakaoti i te {l}{4(2x-y)-7(2y+x)=-36}{-2(x+2)-7y=-18}

$$\left\{ \begin{array} { l } { 4 ( 2 x - y ) - 7 ( 2 y + x ) = - 36 } \\ { - 2 ( x + 2 ) - 7 y = - 18 } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 2x-y.

$$8x-4y-7\left(2y+x\right)=-36$$

Whakamahia te āhuatanga tohatoha hei whakarea te -7 ki te 2y+x.

$$8x-4y-14y-7x=-36$$

Pahekotia te -4y me -14y, ka -18y.

$$8x-18y-7x=-36$$

Pahekotia te 8x me -7x, ka x.

$$x-18y=-36$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te x+2.

$$-2x-4-7y=-18$$

Me tāpiri te 4 ki ngā taha e rua.

$$-2x-7y=-18+4$$

Tāpirihia te -18 ki te 4, ka -14.

$$-2x-7y=-14$$

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.

$$x-18y=-36,-2x-7y=-14$$

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

$$x-18y=-36$$

Me tāpiri 18y ki ngā taha e rua o te whārite.

$$x=18y-36$$

Whakakapia te -36+18y mō te x ki tērā atu whārite, -2x-7y=-14.

$$-2\left(18y-36\right)-7y=-14$$

Whakareatia -2 ki te -36+18y.

$$-36y+72-7y=-14$$

Tāpiri -36y ki te -7y.

$$-43y+72=-14$$

Me tango 72 mai i ngā taha e rua o te whārite.

$$-43y=-86$$

Whakawehea ngā taha e rua ki te -43.

$$y=2$$

Whakaurua te 2 mō y ki x=18y-36. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=18\times 2-36$$

Whakareatia 18 ki te 2.

$$x=36-36$$

Tāpiri -36 ki te 36.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 2x-y.

$$8x-4y-7\left(2y+x\right)=-36$$

Whakamahia te āhuatanga tohatoha hei whakarea te -7 ki te 2y+x.

$$8x-4y-14y-7x=-36$$

Pahekotia te -4y me -14y, ka -18y.

$$8x-18y-7x=-36$$

Pahekotia te 8x me -7x, ka x.

$$x-18y=-36$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te x+2.

$$-2x-4-7y=-18$$

Me tāpiri te 4 ki ngā taha e rua.

$$-2x-7y=-18+4$$

Tāpirihia te -18 ki te 4, ka -14.

$$-2x-7y=-14$$

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

$$x-18y=-36,-2x-7y=-14$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-18\\-2&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-36\\-14\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-18\\-2&-7\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-18\\-2&-7\end{matrix}\right))\left(\begin{matrix}1&-18\\-2&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-18\\-2&-7\end{matrix}\right))\left(\begin{matrix}-36\\-14\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-18\\-2&-7\end{matrix}\right))\left(\begin{matrix}-36\\-14\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-18\\-2&-7\end{matrix}\right))\left(\begin{matrix}-36\\-14\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{-7-\left(-18\left(-2\right)\right)}&-\frac{-18}{-7-\left(-18\left(-2\right)\right)}\\-\frac{-2}{-7-\left(-18\left(-2\right)\right)}&\frac{1}{-7-\left(-18\left(-2\right)\right)}\end{matrix}\right)\left(\begin{matrix}-36\\-14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{43}&-\frac{18}{43}\\-\frac{2}{43}&-\frac{1}{43}\end{matrix}\right)\left(\begin{matrix}-36\\-14\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{43}\left(-36\right)-\frac{18}{43}\left(-14\right)\\-\frac{2}{43}\left(-36\right)-\frac{1}{43}\left(-14\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=0,y=2$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 2x-y.

$$8x-4y-7\left(2y+x\right)=-36$$

Whakamahia te āhuatanga tohatoha hei whakarea te -7 ki te 2y+x.

$$8x-4y-14y-7x=-36$$

Pahekotia te -4y me -14y, ka -18y.

$$8x-18y-7x=-36$$

Pahekotia te 8x me -7x, ka x.

$$x-18y=-36$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te x+2.

$$-2x-4-7y=-18$$

Me tāpiri te 4 ki ngā taha e rua.

$$-2x-7y=-18+4$$

Tāpirihia te -18 ki te 4, ka -14.

$$-2x-7y=-14$$

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.

$$x-18y=-36,-2x-7y=-14$$

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.

$$-2x-2\left(-18\right)y=-2\left(-36\right),-2x-7y=-14$$

Whakarūnātia.

$$-2x+36y=72,-2x-7y=-14$$

Me tango -2x-7y=-14 mai i -2x+36y=72 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x+36y+7y=72+14$$

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.

$$36y+7y=72+14$$

Tāpiri 36y ki te 7y.

$$43y=72+14$$

Tāpiri 72 ki te 14.

$$43y=86$$

Whakawehea ngā taha e rua ki te 43.

$$y=2$$

Whakaurua te 2 mō y ki -2x-7y=-14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x-7\times 2=-14$$

Whakareatia -7 ki te 2.

$$-2x-14=-14$$

Me tāpiri 14 ki ngā taha e rua o te whārite.

$$-2x=0$$

Whakawehea ngā taha e rua ki te -2.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=2$$