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

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

Whakaoti mō x, y

$$x=7$$

$$y=10$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

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

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

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

Tangohia te 2y mai i ngā taha e rua.

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

Me tāpiri te 3 ki ngā taha e rua.

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

Tāpirihia te -2 ki te 3, ka 1.

$$3x-2y=1$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te y-1.

$$4y-4=3\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+5.

$$4y-4=3x+15$$

Tangohia te 3x mai i ngā taha e rua.

$$4y-4-3x=15$$

Me tāpiri te 4 ki ngā taha e rua.

$$4y-3x=15+4$$

Tāpirihia te 15 ki te 4, ka 19.

$$4y-3x=19$$

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.

$$3x-2y=1,-3x+4y=19$$

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+4y=19.

$$-3\left(\frac{2}{3}y+\frac{1}{3}\right)+4y=19$$

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

$$-2y-1+4y=19$$

Tāpiri -2y ki te 4y.

$$2y-1=19$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$2y=20$$

Whakawehea ngā taha e rua ki te 2.

$$y=10$$

Whakaurua te 10 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.

$$x=\frac{2}{3}\times 10+\frac{1}{3}$$

Whakareatia \frac{2}{3} ki te 10.

$$x=\frac{20+1}{3}$$

Tāpiri \frac{1}{3} ki te \frac{20}{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.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=10$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

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

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

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

Tangohia te 2y mai i ngā taha e rua.

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

Me tāpiri te 3 ki ngā taha e rua.

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

Tāpirihia te -2 ki te 3, ka 1.

$$3x-2y=1$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te y-1.

$$4y-4=3\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+5.

$$4y-4=3x+15$$

Tangohia te 3x mai i ngā taha e rua.

$$4y-4-3x=15$$

Me tāpiri te 4 ki ngā taha e rua.

$$4y-3x=15+4$$

Tāpirihia te 15 ki te 4, ka 19.

$$4y-3x=19$$

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

$$3x-2y=1,-3x+4y=19$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-2\\-3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\19\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-2\\-3&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-2\\-3&4\end{matrix}\right))\left(\begin{matrix}3&-2\\-3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\-3&4\end{matrix}\right))\left(\begin{matrix}1\\19\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\-3&4\end{matrix}\right))\left(\begin{matrix}1\\19\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\-3&4\end{matrix}\right))\left(\begin{matrix}1\\19\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{3\times 4-\left(-2\left(-3\right)\right)}&-\frac{-2}{3\times 4-\left(-2\left(-3\right)\right)}\\-\frac{-3}{3\times 4-\left(-2\left(-3\right)\right)}&\frac{3}{3\times 4-\left(-2\left(-3\right)\right)}\end{matrix}\right)\left(\begin{matrix}1\\19\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}&\frac{1}{3}\\\frac{1}{2}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}1\\19\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}+\frac{1}{3}\times 19\\\frac{1}{2}+\frac{1}{2}\times 19\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\10\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=7,y=10$$

Ngā Upane e Whakamahi i te Tangohanga

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

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

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

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

Tangohia te 2y mai i ngā taha e rua.

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

Me tāpiri te 3 ki ngā taha e rua.

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

Tāpirihia te -2 ki te 3, ka 1.

$$3x-2y=1$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te y-1.

$$4y-4=3\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+5.

$$4y-4=3x+15$$

Tangohia te 3x mai i ngā taha e rua.

$$4y-4-3x=15$$

Me tāpiri te 4 ki ngā taha e rua.

$$4y-3x=15+4$$

Tāpirihia te 15 ki te 4, ka 19.

$$4y-3x=19$$

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.

$$3x-2y=1,-3x+4y=19$$

Kia ōrite ai a 3x me -3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$-3\times 3x-3\left(-2\right)y=-3,3\left(-3\right)x+3\times 4y=3\times 19$$

Whakarūnātia.

$$-9x+6y=-3,-9x+12y=57$$

Me tango -9x+12y=57 mai i -9x+6y=-3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-9x+9x+6y-12y=-3-57$$

Tāpiri -9x ki te 9x. Ka whakakore atu ngā kupu -9x me 9x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y-12y=-3-57$$

Tāpiri 6y ki te -12y.

$$-6y=-3-57$$

Tāpiri -3 ki te -57.

$$-6y=-60$$

Whakawehea ngā taha e rua ki te -6.

$$y=10$$

Whakaurua te 10 mō y ki -3x+4y=19. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-3x+4\times 10=19$$

Whakareatia 4 ki te 10.

$$-3x+40=19$$

Me tango 40 mai i ngā taha e rua o te whārite.

$$-3x=-21$$

Whakawehea ngā taha e rua ki te -3.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=10$$