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

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

Whakaoti mō x, y

$$x=-4$$

$$y=-2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x+3.

$$2x+6=3\left(y+1\right)+1$$

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

$$2x+6=3y+3+1$$

Tāpirihia te 3 ki te 1, ka 4.

$$2x+6=3y+4$$

Tangohia te 3y mai i ngā taha e rua.

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

Tangohia te 6 mai i ngā taha e rua.

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

Tangohia te 6 i te 4, ka -2.

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

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

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

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

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

Tāpirihia te -4 ki te 3, ka -1.

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

Tangohia te 2x mai i ngā taha e rua.

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

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

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

Me tāpiri te 3 ki ngā taha e rua.

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

Tāpirihia te -1 ki te 3, ka 2.

$$x-3y=2$$

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.

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

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

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

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

$$2x=3y-2$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

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

$$\frac{3}{2}y-1-3y=2$$

Tāpiri \frac{3y}{2} ki te -3y.

$$-\frac{3}{2}y-1=2$$

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

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

Whakawehea ngā taha e rua o te whārite ki te -\frac{3}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-2$$

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

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

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

$$x=-3-1$$

Tāpiri -1 ki te -3.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=-2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x+3.

$$2x+6=3\left(y+1\right)+1$$

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

$$2x+6=3y+3+1$$

Tāpirihia te 3 ki te 1, ka 4.

$$2x+6=3y+4$$

Tangohia te 3y mai i ngā taha e rua.

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

Tangohia te 6 mai i ngā taha e rua.

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

Tangohia te 6 i te 4, ka -2.

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

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

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

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

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

Tāpirihia te -4 ki te 3, ka -1.

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

Tangohia te 2x mai i ngā taha e rua.

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

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

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

Me tāpiri te 3 ki ngā taha e rua.

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

Tāpirihia te -1 ki te 3, ka 2.

$$x-3y=2$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-4,y=-2$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x+3.

$$2x+6=3\left(y+1\right)+1$$

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

$$2x+6=3y+3+1$$

Tāpirihia te 3 ki te 1, ka 4.

$$2x+6=3y+4$$

Tangohia te 3y mai i ngā taha e rua.

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

Tangohia te 6 mai i ngā taha e rua.

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

Tangohia te 6 i te 4, ka -2.

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

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

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

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

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

Tāpirihia te -4 ki te 3, ka -1.

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

Tangohia te 2x mai i ngā taha e rua.

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

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

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

Me tāpiri te 3 ki ngā taha e rua.

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

Tāpirihia te -1 ki te 3, ka 2.

$$x-3y=2$$

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.

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

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

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

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.

$$2x-x=-2-2$$

Tāpiri 2x ki te -x.

$$x=-2-2$$

Tāpiri -2 ki te -2.

$$x=-4$$

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

$$-4-3y=2$$

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

$$-3y=6$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=-2$$