Whakaoti i te {r}{2(x+1)/3-y=-3}{3(x+5-y)+3x=12}

$$\left. \begin{array} { r } { \frac { 2 ( x + 1 ) } { 3 } - y = - 3 } \\ { 3 ( x + 5 - y ) + 3 x = 12 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 3.

$$2\left(x+1\right)-3y=-9$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x+1.

$$2x+2-3y=-9$$

Tangohia te 2 mai i ngā taha e rua.

$$2x-3y=-9-2$$

Tangohia te 2 i te -9, ka -11.

$$2x-3y=-11$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+5-y.

$$3x+15-3y+3x=12$$

Pahekotia te 3x me 3x, ka 6x.

$$6x+15-3y=12$$

Tangohia te 15 mai i ngā taha e rua.

$$6x-3y=12-15$$

Tangohia te 15 i te 12, ka -3.

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

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.

$$2x-3y=-11,6x-3y=-3$$

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

$$2x-3y=-11$$

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

$$2x=3y-11$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(3y-11\right)$$

Whakareatia \frac{1}{2} ki te 3y-11.

$$x=\frac{3}{2}y-\frac{11}{2}$$

Whakakapia te \frac{3y-11}{2} mō te x ki tērā atu whārite, 6x-3y=-3.

$$6\left(\frac{3}{2}y-\frac{11}{2}\right)-3y=-3$$

Whakareatia 6 ki te \frac{3y-11}{2}.

$$9y-33-3y=-3$$

Tāpiri 9y ki te -3y.

$$6y-33=-3$$

Me tāpiri 33 ki ngā taha e rua o te whārite.

$$6y=30$$

Whakawehea ngā taha e rua ki te 6.

$$y=5$$

Whakaurua te 5 mō y ki x=\frac{3}{2}y-\frac{11}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{2}\times 5-\frac{11}{2}$$

Whakareatia \frac{3}{2} ki te 5.

$$x=\frac{15-11}{2}$$

Tāpiri -\frac{11}{2} ki te \frac{15}{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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=5$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 3.

$$2\left(x+1\right)-3y=-9$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x+1.

$$2x+2-3y=-9$$

Tangohia te 2 mai i ngā taha e rua.

$$2x-3y=-9-2$$

Tangohia te 2 i te -9, ka -11.

$$2x-3y=-11$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+5-y.

$$3x+15-3y+3x=12$$

Pahekotia te 3x me 3x, ka 6x.

$$6x+15-3y=12$$

Tangohia te 15 mai i ngā taha e rua.

$$6x-3y=12-15$$

Tangohia te 15 i te 12, ka -3.

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

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

$$2x-3y=-11,6x-3y=-3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-3\\6&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-11\\-3\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&-3\\6&-3\end{matrix}\right))\left(\begin{matrix}2&-3\\6&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\6&-3\end{matrix}\right))\left(\begin{matrix}-11\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\6&-3\end{matrix}\right))\left(\begin{matrix}-11\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\6&-3\end{matrix}\right))\left(\begin{matrix}-11\\-3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{2\left(-3\right)-\left(-3\times 6\right)}&-\frac{-3}{2\left(-3\right)-\left(-3\times 6\right)}\\-\frac{6}{2\left(-3\right)-\left(-3\times 6\right)}&\frac{2}{2\left(-3\right)-\left(-3\times 6\right)}\end{matrix}\right)\left(\begin{matrix}-11\\-3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&\frac{1}{4}\\-\frac{1}{2}&\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}-11\\-3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\left(-11\right)+\frac{1}{4}\left(-3\right)\\-\frac{1}{2}\left(-11\right)+\frac{1}{6}\left(-3\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=5$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 3.

$$2\left(x+1\right)-3y=-9$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x+1.

$$2x+2-3y=-9$$

Tangohia te 2 mai i ngā taha e rua.

$$2x-3y=-9-2$$

Tangohia te 2 i te -9, ka -11.

$$2x-3y=-11$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+5-y.

$$3x+15-3y+3x=12$$

Pahekotia te 3x me 3x, ka 6x.

$$6x+15-3y=12$$

Tangohia te 15 mai i ngā taha e rua.

$$6x-3y=12-15$$

Tangohia te 15 i te 12, ka -3.

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

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.

$$2x-3y=-11,6x-3y=-3$$

Me tango 6x-3y=-3 mai i 2x-3y=-11 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-6x-3y+3y=-11+3$$

Tāpiri -3y ki te 3y. Ka whakakore atu ngā kupu -3y me 3y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2x-6x=-11+3$$

Tāpiri 2x ki te -6x.

$$-4x=-11+3$$

Tāpiri -11 ki te 3.

$$-4x=-8$$

Whakawehea ngā taha e rua ki te -4.

$$x=2$$

Whakaurua te 2 mō x ki 6x-3y=-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$6\times 2-3y=-3$$

Whakareatia 6 ki te 2.

$$12-3y=-3$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$-3y=-15$$

Whakawehea ngā taha e rua ki te -3.

$$y=5$$

Kua oti te pūnaha te whakatau.

$$x=2,y=5$$