Whakaoti i te {l}{3x-1/2-4y-7/3=2}{3y-6/4-5-x/6=-1frac{5}{12}}

$$\left\{ \begin{array} { l } { \frac { 3 x - 1 } { 2 } - \frac { 4 y - 7 } { 3 } = 2 } \\ { \frac { 3 y - 6 } { 4 } - \frac { 5 - x } { 6 } = - 1 \frac { 5 } { 12 } } \end{array} \right.$$

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 6, arā, te tauraro pātahi he tino iti rawa te kitea o 2,3.

$$3\left(3x-1\right)-2\left(4y-7\right)=12$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 3x-1.

$$9x-3-2\left(4y-7\right)=12$$

Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te 4y-7.

$$9x-3-8y+14=12$$

Tāpirihia te -3 ki te 14, ka 11.

$$9x+11-8y=12$$

Tangohia te 11 mai i ngā taha e rua.

$$9x-8y=12-11$$

Tangohia te 11 i te 12, ka 1.

$$9x-8y=1$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 4,6,12.

$$3\left(3y-6\right)-2\left(5-x\right)=-\left(1\times 12+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 3y-6.

$$9y-18-2\left(5-x\right)=-\left(1\times 12+5\right)$$

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

$$9y-18-10+2x=-\left(1\times 12+5\right)$$

Tangohia te 10 i te -18, ka -28.

$$9y-28+2x=-\left(1\times 12+5\right)$$

Whakareatia te 1 ki te 12, ka 12.

$$9y-28+2x=-\left(12+5\right)$$

Tāpirihia te 12 ki te 5, ka 17.

$$9y-28+2x=-17$$

Me tāpiri te 28 ki ngā taha e rua.

$$9y+2x=-17+28$$

Tāpirihia te -17 ki te 28, ka 11.

$$9y+2x=11$$

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.

$$9x-8y=1,2x+9y=11$$

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

$$9x-8y=1$$

Me tāpiri 8y ki ngā taha e rua o te whārite.

$$9x=8y+1$$

Whakawehea ngā taha e rua ki te 9.

$$x=\frac{1}{9}\left(8y+1\right)$$

Whakareatia \frac{1}{9} ki te 8y+1.

$$x=\frac{8}{9}y+\frac{1}{9}$$

Whakakapia te \frac{8y+1}{9} mō te x ki tērā atu whārite, 2x+9y=11.

$$2\left(\frac{8}{9}y+\frac{1}{9}\right)+9y=11$$

Whakareatia 2 ki te \frac{8y+1}{9}.

$$\frac{16}{9}y+\frac{2}{9}+9y=11$$

Tāpiri \frac{16y}{9} ki te 9y.

$$\frac{97}{9}y+\frac{2}{9}=11$$

Me tango \frac{2}{9} mai i ngā taha e rua o te whārite.

$$\frac{97}{9}y=\frac{97}{9}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{97}{9}, 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{8}{9}y+\frac{1}{9}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{8+1}{9}$$

Tāpiri \frac{1}{9} ki te \frac{8}{9} 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. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 2,3.

$$3\left(3x-1\right)-2\left(4y-7\right)=12$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 3x-1.

$$9x-3-2\left(4y-7\right)=12$$

Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te 4y-7.

$$9x-3-8y+14=12$$

Tāpirihia te -3 ki te 14, ka 11.

$$9x+11-8y=12$$

Tangohia te 11 mai i ngā taha e rua.

$$9x-8y=12-11$$

Tangohia te 11 i te 12, ka 1.

$$9x-8y=1$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 4,6,12.

$$3\left(3y-6\right)-2\left(5-x\right)=-\left(1\times 12+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 3y-6.

$$9y-18-2\left(5-x\right)=-\left(1\times 12+5\right)$$

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

$$9y-18-10+2x=-\left(1\times 12+5\right)$$

Tangohia te 10 i te -18, ka -28.

$$9y-28+2x=-\left(1\times 12+5\right)$$

Whakareatia te 1 ki te 12, ka 12.

$$9y-28+2x=-\left(12+5\right)$$

Tāpirihia te 12 ki te 5, ka 17.

$$9y-28+2x=-17$$

Me tāpiri te 28 ki ngā taha e rua.

$$9y+2x=-17+28$$

Tāpirihia te -17 ki te 28, ka 11.

$$9y+2x=11$$

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

$$9x-8y=1,2x+9y=11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}9&-8\\2&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\11\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}9&-8\\2&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&-8\\2&9\end{matrix}\right))\left(\begin{matrix}9&-8\\2&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-8\\2&9\end{matrix}\right))\left(\begin{matrix}1\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-8\\2&9\end{matrix}\right))\left(\begin{matrix}1\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-8\\2&9\end{matrix}\right))\left(\begin{matrix}1\\11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{9\times 9-\left(-8\times 2\right)}&-\frac{-8}{9\times 9-\left(-8\times 2\right)}\\-\frac{2}{9\times 9-\left(-8\times 2\right)}&\frac{9}{9\times 9-\left(-8\times 2\right)}\end{matrix}\right)\left(\begin{matrix}1\\11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{97}&\frac{8}{97}\\-\frac{2}{97}&\frac{9}{97}\end{matrix}\right)\left(\begin{matrix}1\\11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{97}+\frac{8}{97}\times 11\\-\frac{2}{97}+\frac{9}{97}\times 11\end{matrix}\right)$$

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 6, arā, te tauraro pātahi he tino iti rawa te kitea o 2,3.

$$3\left(3x-1\right)-2\left(4y-7\right)=12$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 3x-1.

$$9x-3-2\left(4y-7\right)=12$$

Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te 4y-7.

$$9x-3-8y+14=12$$

Tāpirihia te -3 ki te 14, ka 11.

$$9x+11-8y=12$$

Tangohia te 11 mai i ngā taha e rua.

$$9x-8y=12-11$$

Tangohia te 11 i te 12, ka 1.

$$9x-8y=1$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 4,6,12.

$$3\left(3y-6\right)-2\left(5-x\right)=-\left(1\times 12+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 3y-6.

$$9y-18-2\left(5-x\right)=-\left(1\times 12+5\right)$$

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

$$9y-18-10+2x=-\left(1\times 12+5\right)$$

Tangohia te 10 i te -18, ka -28.

$$9y-28+2x=-\left(1\times 12+5\right)$$

Whakareatia te 1 ki te 12, ka 12.

$$9y-28+2x=-\left(12+5\right)$$

Tāpirihia te 12 ki te 5, ka 17.

$$9y-28+2x=-17$$

Me tāpiri te 28 ki ngā taha e rua.

$$9y+2x=-17+28$$

Tāpirihia te -17 ki te 28, ka 11.

$$9y+2x=11$$

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.

$$9x-8y=1,2x+9y=11$$

Kia ōrite ai a 9x 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 9.

$$2\times 9x+2\left(-8\right)y=2,9\times 2x+9\times 9y=9\times 11$$

Whakarūnātia.

$$18x-16y=2,18x+81y=99$$

Me tango 18x+81y=99 mai i 18x-16y=2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$18x-18x-16y-81y=2-99$$

Tāpiri 18x ki te -18x. Ka whakakore atu ngā kupu 18x me -18x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-16y-81y=2-99$$

Tāpiri -16y ki te -81y.

$$-97y=2-99$$

Tāpiri 2 ki te -99.

$$-97y=-97$$

Whakawehea ngā taha e rua ki te -97.

$$y=1$$

Whakaurua te 1 mō y ki 2x+9y=11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+9=11$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$2x=2$$

Whakawehea ngā taha e rua ki te 2.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=1$$