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

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

Whakaoti mō x, y

$$x=3$$

$$y=4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

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

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

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

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

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

Tangohia te 2y mai i ngā taha e rua.

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

Tangohia te 3 mai i ngā taha e rua.

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

Tangohia te 3 i te 4, ka 1.

$$3x-2y=1$$

Whakaarohia te whārite tuarua. Tē taea kia ōrite te tāupe y ki 1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 3\left(y-1\right), arā, te tauraro pātahi he tino iti rawa te kitea o y-1,3.

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

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

$$3x-6=y-1$$

Tangohia te y mai i ngā taha e rua.

$$3x-6-y=-1$$

Me tāpiri te 6 ki ngā taha e rua.

$$3x-y=-1+6$$

Tāpirihia te -1 ki te 6, ka 5.

$$3x-y=5$$

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.

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

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

$$3x-2y=1$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$3x=2y+1$$

Whakawehea ngā taha e rua ki te 3.

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

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

$$x=\frac{2}{3}y+\frac{1}{3}$$

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

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

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

$$2y+1-y=5$$

Tāpiri 2y ki te -y.

$$y+1=5$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$y=4$$

Whakaurua te 4 mō y ki x=\frac{2}{3}y+\frac{1}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{3}\times 4+\frac{1}{3}$$

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

$$x=\frac{8+1}{3}$$

Tāpiri \frac{1}{3} ki te \frac{8}{3} 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.

$$x=3,y=4$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

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

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

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

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

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

Tangohia te 2y mai i ngā taha e rua.

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

Tangohia te 3 mai i ngā taha e rua.

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

Tangohia te 3 i te 4, ka 1.

$$3x-2y=1$$

Whakaarohia te whārite tuarua. Tē taea kia ōrite te tāupe y ki 1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 3\left(y-1\right), arā, te tauraro pātahi he tino iti rawa te kitea o y-1,3.

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

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

$$3x-6=y-1$$

Tangohia te y mai i ngā taha e rua.

$$3x-6-y=-1$$

Me tāpiri te 6 ki ngā taha e rua.

$$3x-y=-1+6$$

Tāpirihia te -1 ki te 6, ka 5.

$$3x-y=5$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=4$$

Ngā Upane e Whakamahi i te Tangohanga

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

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

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

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

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

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

Tangohia te 2y mai i ngā taha e rua.

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

Tangohia te 3 mai i ngā taha e rua.

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

Tangohia te 3 i te 4, ka 1.

$$3x-2y=1$$

Whakaarohia te whārite tuarua. Tē taea kia ōrite te tāupe y ki 1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 3\left(y-1\right), arā, te tauraro pātahi he tino iti rawa te kitea o y-1,3.

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

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

$$3x-6=y-1$$

Tangohia te y mai i ngā taha e rua.

$$3x-6-y=-1$$

Me tāpiri te 6 ki ngā taha e rua.

$$3x-y=-1+6$$

Tāpirihia te -1 ki te 6, ka 5.

$$3x-y=5$$

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.

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

Me tango 3x-y=5 mai i 3x-2y=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x-2y+y=1-5$$

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

$$-2y+y=1-5$$

Tāpiri -2y ki te y.

$$-y=1-5$$

Tāpiri 1 ki te -5.

$$-y=-4$$

Whakawehea ngā taha e rua ki te -1.

$$y=4$$

Whakaurua te 4 mō y ki 3x-y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-4=5$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$3x=9$$

Whakawehea ngā taha e rua ki te 3.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=4$$