Whakaoti i te {l}{x-y/5-y/2=x-1}{x/3+y+2/2=1}

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

Whakaoti mō x, y

$$x=3$$

$$y=-2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 10, arā, te tauraro pātahi he tino iti rawa te kitea o 5,2.

$$2\left(x-y\right)-5y=10x-10$$

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

$$2x-2y-5y=10x-10$$

Pahekotia te -2y me -5y, ka -7y.

$$2x-7y=10x-10$$

Tangohia te 10x mai i ngā taha e rua.

$$2x-7y-10x=-10$$

Pahekotia te 2x me -10x, ka -8x.

$$-8x-7y=-10$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2.

$$2x+3\left(y+2\right)=6$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te y+2.

$$2x+3y+6=6$$

Tangohia te 6 mai i ngā taha e rua.

$$2x+3y=6-6$$

Tangohia te 6 i te 6, ka 0.

$$2x+3y=0$$

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.

$$-8x-7y=-10,2x+3y=0$$

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

$$-8x-7y=-10$$

Me tāpiri 7y ki ngā taha e rua o te whārite.

$$-8x=7y-10$$

Whakawehea ngā taha e rua ki te -8.

$$x=-\frac{1}{8}\left(7y-10\right)$$

Whakareatia -\frac{1}{8} ki te 7y-10.

$$x=-\frac{7}{8}y+\frac{5}{4}$$

Whakakapia te -\frac{7y}{8}+\frac{5}{4} mō te x ki tērā atu whārite, 2x+3y=0.

$$2\left(-\frac{7}{8}y+\frac{5}{4}\right)+3y=0$$

Whakareatia 2 ki te -\frac{7y}{8}+\frac{5}{4}.

$$-\frac{7}{4}y+\frac{5}{2}+3y=0$$

Tāpiri -\frac{7y}{4} ki te 3y.

$$\frac{5}{4}y+\frac{5}{2}=0$$

Me tango \frac{5}{2} mai i ngā taha e rua o te whārite.

$$\frac{5}{4}y=-\frac{5}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-2$$

Whakaurua te -2 mō y ki x=-\frac{7}{8}y+\frac{5}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{7}{8}\left(-2\right)+\frac{5}{4}$$

Whakareatia -\frac{7}{8} ki te -2.

$$x=\frac{7+5}{4}$$

Tāpiri \frac{5}{4} ki te \frac{7}{4} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=-2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 10, arā, te tauraro pātahi he tino iti rawa te kitea o 5,2.

$$2\left(x-y\right)-5y=10x-10$$

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

$$2x-2y-5y=10x-10$$

Pahekotia te -2y me -5y, ka -7y.

$$2x-7y=10x-10$$

Tangohia te 10x mai i ngā taha e rua.

$$2x-7y-10x=-10$$

Pahekotia te 2x me -10x, ka -8x.

$$-8x-7y=-10$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2.

$$2x+3\left(y+2\right)=6$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te y+2.

$$2x+3y+6=6$$

Tangohia te 6 mai i ngā taha e rua.

$$2x+3y=6-6$$

Tangohia te 6 i te 6, ka 0.

$$2x+3y=0$$

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

$$-8x-7y=-10,2x+3y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-8&-7\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-10\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-8&-7\\2&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}-8&-7\\2&3\end{matrix}\right))\left(\begin{matrix}-8&-7\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-8&-7\\2&3\end{matrix}\right))\left(\begin{matrix}-10\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-8&-7\\2&3\end{matrix}\right))\left(\begin{matrix}-10\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-8&-7\\2&3\end{matrix}\right))\left(\begin{matrix}-10\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{-8\times 3-\left(-7\times 2\right)}&-\frac{-7}{-8\times 3-\left(-7\times 2\right)}\\-\frac{2}{-8\times 3-\left(-7\times 2\right)}&-\frac{8}{-8\times 3-\left(-7\times 2\right)}\end{matrix}\right)\left(\begin{matrix}-10\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{10}&-\frac{7}{10}\\\frac{1}{5}&\frac{4}{5}\end{matrix}\right)\left(\begin{matrix}-10\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{10}\left(-10\right)\\\frac{1}{5}\left(-10\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=-2$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 10, arā, te tauraro pātahi he tino iti rawa te kitea o 5,2.

$$2\left(x-y\right)-5y=10x-10$$

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

$$2x-2y-5y=10x-10$$

Pahekotia te -2y me -5y, ka -7y.

$$2x-7y=10x-10$$

Tangohia te 10x mai i ngā taha e rua.

$$2x-7y-10x=-10$$

Pahekotia te 2x me -10x, ka -8x.

$$-8x-7y=-10$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2.

$$2x+3\left(y+2\right)=6$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te y+2.

$$2x+3y+6=6$$

Tangohia te 6 mai i ngā taha e rua.

$$2x+3y=6-6$$

Tangohia te 6 i te 6, ka 0.

$$2x+3y=0$$

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.

$$-8x-7y=-10,2x+3y=0$$

Kia ōrite ai a -8x 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 -8.

$$2\left(-8\right)x+2\left(-7\right)y=2\left(-10\right),-8\times 2x-8\times 3y=0$$

Whakarūnātia.

$$-16x-14y=-20,-16x-24y=0$$

Me tango -16x-24y=0 mai i -16x-14y=-20 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-16x+16x-14y+24y=-20$$

Tāpiri -16x ki te 16x. Ka whakakore atu ngā kupu -16x me 16x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-14y+24y=-20$$

Tāpiri -14y ki te 24y.

$$10y=-20$$

Whakawehea ngā taha e rua ki te 10.

$$y=-2$$

Whakaurua te -2 mō y ki 2x+3y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+3\left(-2\right)=0$$

Whakareatia 3 ki te -2.

$$2x-6=0$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$2x=6$$

Whakawehea ngā taha e rua ki te 2.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=-2$$