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

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

Whakaoti mō x, y

$$x=-1$$

$$y=1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tē taea kia ōrite te tāupe y ki \frac{2}{3} nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 3y-2.

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

Tangohia te 3y mai i ngā taha e rua.

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

Tangohia te 3 mai i ngā taha e rua.

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

Tangohia te 3 i te -2, ka -5.

$$2x-3y=-5$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te x ki te 2y-5.

$$2xy-5x-2y\left(x+3\right)=2x+1$$

Tangohia te 2x mai i ngā taha e rua.

$$2xy-5x-2y\left(x+3\right)-2x=1$$

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

$$2xy-5x-2yx-6y-2x=1$$

Pahekotia te 2xy me -2yx, ka 0.

$$-5x-6y-2x=1$$

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

$$-7x-6y=1$$

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.

$$2x-3y=-5,-7x-6y=1$$

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

$$2x-3y=-5$$

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

$$2x=3y-5$$

Whakawehea ngā taha e rua ki te 2.

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

Whakareatia \frac{1}{2} ki te 3y-5.

$$x=\frac{3}{2}y-\frac{5}{2}$$

Whakakapia te \frac{3y-5}{2} mō te x ki tērā atu whārite, -7x-6y=1.

$$-7\left(\frac{3}{2}y-\frac{5}{2}\right)-6y=1$$

Whakareatia -7 ki te \frac{3y-5}{2}.

$$-\frac{21}{2}y+\frac{35}{2}-6y=1$$

Tāpiri -\frac{21y}{2} ki te -6y.

$$-\frac{33}{2}y+\frac{35}{2}=1$$

Me tango \frac{35}{2} mai i ngā taha e rua o te whārite.

$$-\frac{33}{2}y=-\frac{33}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{33}{2}, 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=\frac{3}{2}y-\frac{5}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3-5}{2}$$

Tāpiri -\frac{5}{2} ki te \frac{3}{2} 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.

$$x=-1$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tē taea kia ōrite te tāupe y ki \frac{2}{3} nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 3y-2.

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

Tangohia te 3y mai i ngā taha e rua.

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

Tangohia te 3 mai i ngā taha e rua.

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

Tangohia te 3 i te -2, ka -5.

$$2x-3y=-5$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te x ki te 2y-5.

$$2xy-5x-2y\left(x+3\right)=2x+1$$

Tangohia te 2x mai i ngā taha e rua.

$$2xy-5x-2y\left(x+3\right)-2x=1$$

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

$$2xy-5x-2yx-6y-2x=1$$

Pahekotia te 2xy me -2yx, ka 0.

$$-5x-6y-2x=1$$

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

$$-7x-6y=1$$

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

$$2x-3y=-5,-7x-6y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-3\\-7&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\\1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-3\\-7&-6\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-3\\-7&-6\end{matrix}\right))\left(\begin{matrix}2&-3\\-7&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\-7&-6\end{matrix}\right))\left(\begin{matrix}-5\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\-7&-6\end{matrix}\right))\left(\begin{matrix}-5\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\-7&-6\end{matrix}\right))\left(\begin{matrix}-5\\1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{2\left(-6\right)-\left(-3\left(-7\right)\right)}&-\frac{-3}{2\left(-6\right)-\left(-3\left(-7\right)\right)}\\-\frac{-7}{2\left(-6\right)-\left(-3\left(-7\right)\right)}&\frac{2}{2\left(-6\right)-\left(-3\left(-7\right)\right)}\end{matrix}\right)\left(\begin{matrix}-5\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}&-\frac{1}{11}\\-\frac{7}{33}&-\frac{2}{33}\end{matrix}\right)\left(\begin{matrix}-5\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}\left(-5\right)-\frac{1}{11}\\-\frac{7}{33}\left(-5\right)-\frac{2}{33}\end{matrix}\right)$$

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. Tē taea kia ōrite te tāupe y ki \frac{2}{3} nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 3y-2.

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

Tangohia te 3y mai i ngā taha e rua.

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

Tangohia te 3 mai i ngā taha e rua.

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

Tangohia te 3 i te -2, ka -5.

$$2x-3y=-5$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te x ki te 2y-5.

$$2xy-5x-2y\left(x+3\right)=2x+1$$

Tangohia te 2x mai i ngā taha e rua.

$$2xy-5x-2y\left(x+3\right)-2x=1$$

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

$$2xy-5x-2yx-6y-2x=1$$

Pahekotia te 2xy me -2yx, ka 0.

$$-5x-6y-2x=1$$

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

$$-7x-6y=1$$

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.

$$2x-3y=-5,-7x-6y=1$$

Kia ōrite ai a 2x me -7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$-7\times 2x-7\left(-3\right)y=-7\left(-5\right),2\left(-7\right)x+2\left(-6\right)y=2$$

Whakarūnātia.

$$-14x+21y=35,-14x-12y=2$$

Me tango -14x-12y=2 mai i -14x+21y=35 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-14x+14x+21y+12y=35-2$$

Tāpiri -14x ki te 14x. Ka whakakore atu ngā kupu -14x me 14x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$21y+12y=35-2$$

Tāpiri 21y ki te 12y.

$$33y=35-2$$

Tāpiri 35 ki te -2.

$$33y=33$$

Whakawehea ngā taha e rua ki te 33.

$$y=1$$

Whakaurua te 1 mō y ki -7x-6y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-7x-6=1$$

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

$$-7x=7$$

Whakawehea ngā taha e rua ki te -7.

$$x=-1$$

Kua oti te pūnaha te whakatau.

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