Whakaoti i te {l}{y+1/3x-4=1/2}{5x+y/3x+11=1}

$$\left. \begin{array} { l } { \frac { y + 1 } { 3 x - 4 } = \frac { 1 } { 2 } } \\ { \frac { 5 x + y } { 3 x + 11 } = 1 } \end{array} \right.$$

Whakaoti mō y, x

$$x=4$$

$$y=3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tē taea kia ōrite te tāupe x ki \frac{4}{3} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 2\left(3x-4\right), arā, te tauraro pātahi he tino iti rawa te kitea o 3x-4,2.

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

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

$$2y+2=3x-4$$

Tangohia te 3x mai i ngā taha e rua.

$$2y+2-3x=-4$$

Tangohia te 2 mai i ngā taha e rua.

$$2y-3x=-4-2$$

Tangohia te 2 i te -4, ka -6.

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

Whakaarohia te whārite tuarua. Tē taea kia ōrite te tāupe x ki -\frac{11}{3} nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 3x+11.

$$5x+y=3x+11$$

Tangohia te 3x mai i ngā taha e rua.

$$5x+y-3x=11$$

Pahekotia te 5x me -3x, ka 2x.

$$2x+y=11$$

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.

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

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

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

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

$$2y=3x-6$$

Whakawehea ngā taha e rua ki te 2.

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

Whakareatia \frac{1}{2} ki te -6+3x.

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

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

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

Tāpiri \frac{3x}{2} ki te 2x.

$$\frac{7}{2}x-3=11$$

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

$$\frac{7}{2}x=14$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=4$$

Whakaurua te 4 mō x ki y=\frac{3}{2}x-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=\frac{3}{2}\times 4-3$$

Whakareatia \frac{3}{2} ki te 4.

$$y=6-3$$

Tāpiri -3 ki te 6.

$$y=3$$

Kua oti te pūnaha te whakatau.

$$y=3,x=4$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tē taea kia ōrite te tāupe x ki \frac{4}{3} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 2\left(3x-4\right), arā, te tauraro pātahi he tino iti rawa te kitea o 3x-4,2.

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

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

$$2y+2=3x-4$$

Tangohia te 3x mai i ngā taha e rua.

$$2y+2-3x=-4$$

Tangohia te 2 mai i ngā taha e rua.

$$2y-3x=-4-2$$

Tangohia te 2 i te -4, ka -6.

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

Whakaarohia te whārite tuarua. Tē taea kia ōrite te tāupe x ki -\frac{11}{3} nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 3x+11.

$$5x+y=3x+11$$

Tangohia te 3x mai i ngā taha e rua.

$$5x+y-3x=11$$

Pahekotia te 5x me -3x, ka 2x.

$$2x+y=11$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-3\\1&2\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}&\frac{3}{7}\\-\frac{1}{7}&\frac{2}{7}\end{matrix}\right)\left(\begin{matrix}-6\\11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}\left(-6\right)+\frac{3}{7}\times 11\\-\frac{1}{7}\left(-6\right)+\frac{2}{7}\times 11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}3\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=3,x=4$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tē taea kia ōrite te tāupe x ki \frac{4}{3} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 2\left(3x-4\right), arā, te tauraro pātahi he tino iti rawa te kitea o 3x-4,2.

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

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

$$2y+2=3x-4$$

Tangohia te 3x mai i ngā taha e rua.

$$2y+2-3x=-4$$

Tangohia te 2 mai i ngā taha e rua.

$$2y-3x=-4-2$$

Tangohia te 2 i te -4, ka -6.

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

Whakaarohia te whārite tuarua. Tē taea kia ōrite te tāupe x ki -\frac{11}{3} nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 3x+11.

$$5x+y=3x+11$$

Tangohia te 3x mai i ngā taha e rua.

$$5x+y-3x=11$$

Pahekotia te 5x me -3x, ka 2x.

$$2x+y=11$$

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.

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

Kia ōrite ai a 2y me y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$2y-3x=-6,2y+2\times 2x=2\times 11$$

Whakarūnātia.

$$2y-3x=-6,2y+4x=22$$

Me tango 2y+4x=22 mai i 2y-3x=-6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2y-2y-3x-4x=-6-22$$

Tāpiri 2y ki te -2y. Ka whakakore atu ngā kupu 2y me -2y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-3x-4x=-6-22$$

Tāpiri -3x ki te -4x.

$$-7x=-6-22$$

Tāpiri -6 ki te -22.

$$-7x=-28$$

Whakawehea ngā taha e rua ki te -7.

$$x=4$$

Whakaurua te 4 mō x ki y+2x=11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y+2\times 4=11$$

Whakareatia 2 ki te 4.

$$y+8=11$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$y=3$$

Kua oti te pūnaha te whakatau.

$$y=3,x=4$$