Whakaoti i te {l}{1/2x-1+2y/12=2x-5y/3-7/4}{frac{1}{5}x+frac{2}{7}y=-frac{3}{35}}

$$\left. \begin{array} { l } { \frac { 1 } { 2 } x - \frac { 1 + 2 y } { 12 } = \frac { 2 x - 5 y } { 3 } - \frac { 7 } { 4 } } \\ { \frac { 1 } { 5 } x + \frac { 2 } { 7 } y = - \frac { 3 } { 35 } } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=-1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 2,12,3,4.

$$6x-\left(1+2y\right)=4\left(2x-5y\right)-21$$

Hei kimi i te tauaro o 1+2y, kimihia te tauaro o ia taurangi.

$$6x-1-2y=4\left(2x-5y\right)-21$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 2x-5y.

$$6x-1-2y=8x-20y-21$$

Tangohia te 8x mai i ngā taha e rua.

$$6x-1-2y-8x=-20y-21$$

Pahekotia te 6x me -8x, ka -2x.

$$-2x-1-2y=-20y-21$$

Me tāpiri te 20y ki ngā taha e rua.

$$-2x-1-2y+20y=-21$$

Pahekotia te -2y me 20y, ka 18y.

$$-2x-1+18y=-21$$

Me tāpiri te 1 ki ngā taha e rua.

$$-2x+18y=-21+1$$

Tāpirihia te -21 ki te 1, ka -20.

$$-2x+18y=-20$$

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.

$$-2x+18y=-20,\frac{1}{5}x+\frac{2}{7}y=-\frac{3}{35}$$

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

$$-2x+18y=-20$$

Me tango 18y mai i ngā taha e rua o te whārite.

$$-2x=-18y-20$$

Whakawehea ngā taha e rua ki te -2.

$$x=-\frac{1}{2}\left(-18y-20\right)$$

Whakareatia -\frac{1}{2} ki te -18y-20.

$$x=9y+10$$

Whakakapia te 9y+10 mō te x ki tērā atu whārite, \frac{1}{5}x+\frac{2}{7}y=-\frac{3}{35}.

$$\frac{1}{5}\left(9y+10\right)+\frac{2}{7}y=-\frac{3}{35}$$

Whakareatia \frac{1}{5} ki te 9y+10.

$$\frac{9}{5}y+2+\frac{2}{7}y=-\frac{3}{35}$$

Tāpiri \frac{9y}{5} ki te \frac{2y}{7}.

$$\frac{73}{35}y+2=-\frac{3}{35}$$

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

$$\frac{73}{35}y=-\frac{73}{35}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{73}{35}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-1$$

Whakaurua te -1 mō y ki x=9y+10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=9\left(-1\right)+10$$

Whakareatia 9 ki te -1.

$$x=-9+10$$

Tāpiri 10 ki te -9.

$$x=1$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 2,12,3,4.

$$6x-\left(1+2y\right)=4\left(2x-5y\right)-21$$

Hei kimi i te tauaro o 1+2y, kimihia te tauaro o ia taurangi.

$$6x-1-2y=4\left(2x-5y\right)-21$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 2x-5y.

$$6x-1-2y=8x-20y-21$$

Tangohia te 8x mai i ngā taha e rua.

$$6x-1-2y-8x=-20y-21$$

Pahekotia te 6x me -8x, ka -2x.

$$-2x-1-2y=-20y-21$$

Me tāpiri te 20y ki ngā taha e rua.

$$-2x-1-2y+20y=-21$$

Pahekotia te -2y me 20y, ka 18y.

$$-2x-1+18y=-21$$

Me tāpiri te 1 ki ngā taha e rua.

$$-2x+18y=-21+1$$

Tāpirihia te -21 ki te 1, ka -20.

$$-2x+18y=-20$$

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

$$-2x+18y=-20,\frac{1}{5}x+\frac{2}{7}y=-\frac{3}{35}$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-2&18\\\frac{1}{5}&\frac{2}{7}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-20\\-\frac{3}{35}\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-2&18\\\frac{1}{5}&\frac{2}{7}\end{matrix}\right).

$$inverse(\left(\begin{matrix}-2&18\\\frac{1}{5}&\frac{2}{7}\end{matrix}\right))\left(\begin{matrix}-2&18\\\frac{1}{5}&\frac{2}{7}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&18\\\frac{1}{5}&\frac{2}{7}\end{matrix}\right))\left(\begin{matrix}-20\\-\frac{3}{35}\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&18\\\frac{1}{5}&\frac{2}{7}\end{matrix}\right))\left(\begin{matrix}-20\\-\frac{3}{35}\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&18\\\frac{1}{5}&\frac{2}{7}\end{matrix}\right))\left(\begin{matrix}-20\\-\frac{3}{35}\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{2}{7}}{-2\times \frac{2}{7}-18\times \frac{1}{5}}&-\frac{18}{-2\times \frac{2}{7}-18\times \frac{1}{5}}\\-\frac{\frac{1}{5}}{-2\times \frac{2}{7}-18\times \frac{1}{5}}&-\frac{2}{-2\times \frac{2}{7}-18\times \frac{1}{5}}\end{matrix}\right)\left(\begin{matrix}-20\\-\frac{3}{35}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{73}&\frac{315}{73}\\\frac{7}{146}&\frac{35}{73}\end{matrix}\right)\left(\begin{matrix}-20\\-\frac{3}{35}\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{73}\left(-20\right)+\frac{315}{73}\left(-\frac{3}{35}\right)\\\frac{7}{146}\left(-20\right)+\frac{35}{73}\left(-\frac{3}{35}\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 2,12,3,4.

$$6x-\left(1+2y\right)=4\left(2x-5y\right)-21$$

Hei kimi i te tauaro o 1+2y, kimihia te tauaro o ia taurangi.

$$6x-1-2y=4\left(2x-5y\right)-21$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 2x-5y.

$$6x-1-2y=8x-20y-21$$

Tangohia te 8x mai i ngā taha e rua.

$$6x-1-2y-8x=-20y-21$$

Pahekotia te 6x me -8x, ka -2x.

$$-2x-1-2y=-20y-21$$

Me tāpiri te 20y ki ngā taha e rua.

$$-2x-1-2y+20y=-21$$

Pahekotia te -2y me 20y, ka 18y.

$$-2x-1+18y=-21$$

Me tāpiri te 1 ki ngā taha e rua.

$$-2x+18y=-21+1$$

Tāpirihia te -21 ki te 1, ka -20.

$$-2x+18y=-20$$

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.

$$-2x+18y=-20,\frac{1}{5}x+\frac{2}{7}y=-\frac{3}{35}$$

Kia ōrite ai a -2x me \frac{x}{5}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te \frac{1}{5} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -2.

$$\frac{1}{5}\left(-2\right)x+\frac{1}{5}\times 18y=\frac{1}{5}\left(-20\right),-2\times \frac{1}{5}x-2\times \frac{2}{7}y=-2\left(-\frac{3}{35}\right)$$

Whakarūnātia.

$$-\frac{2}{5}x+\frac{18}{5}y=-4,-\frac{2}{5}x-\frac{4}{7}y=\frac{6}{35}$$

Me tango -\frac{2}{5}x-\frac{4}{7}y=\frac{6}{35} mai i -\frac{2}{5}x+\frac{18}{5}y=-4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-\frac{2}{5}x+\frac{2}{5}x+\frac{18}{5}y+\frac{4}{7}y=-4-\frac{6}{35}$$

Tāpiri -\frac{2x}{5} ki te \frac{2x}{5}. Ka whakakore atu ngā kupu -\frac{2x}{5} me \frac{2x}{5}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\frac{18}{5}y+\frac{4}{7}y=-4-\frac{6}{35}$$

Tāpiri \frac{18y}{5} ki te \frac{4y}{7}.

$$\frac{146}{35}y=-4-\frac{6}{35}$$

Tāpiri -4 ki te -\frac{6}{35}.

$$\frac{146}{35}y=-\frac{146}{35}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{146}{35}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-1$$

Whakaurua te -1 mō y ki \frac{1}{5}x+\frac{2}{7}y=-\frac{3}{35}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$\frac{1}{5}x+\frac{2}{7}\left(-1\right)=-\frac{3}{35}$$

Whakareatia \frac{2}{7} ki te -1.

$$\frac{1}{5}x-\frac{2}{7}=-\frac{3}{35}$$

Me tāpiri \frac{2}{7} ki ngā taha e rua o te whārite.

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

Me whakarea ngā taha e rua ki te 5.

$$x=1$$

Kua oti te pūnaha te whakatau.

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