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

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

Whakaoti mō x, y

$$x=1$$

$$y=\frac{2}{3}\approx 0.666666667$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

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

Hei kimi i te tauaro o x-y, kimihia te tauaro o ia taurangi.

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

Pahekotia te 2x me -x, ka x.

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

Pahekotia te 2y me y, ka 3y.

$$x+3y=3$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te x-y.

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

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

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

Pahekotia te y me 2y, ka 3y.

$$-x+3y=1$$

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.

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

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

$$x+3y=3$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$x=-3y+3$$

Whakakapia te -3y+3 mō te x ki tērā atu whārite, -x+3y=1.

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

Whakareatia -1 ki te -3y+3.

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

Tāpiri 3y ki te 3y.

$$6y-3=1$$

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

$$6y=4$$

Whakawehea ngā taha e rua ki te 6.

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

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

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

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

$$x=-2+3$$

Tāpiri 3 ki te -2.

$$x=1$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

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

Hei kimi i te tauaro o x-y, kimihia te tauaro o ia taurangi.

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

Pahekotia te 2x me -x, ka x.

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

Pahekotia te 2y me y, ka 3y.

$$x+3y=3$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te x-y.

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

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

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

Pahekotia te y me 2y, ka 3y.

$$-x+3y=1$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

Ngā Upane e Whakamahi i te Tangohanga

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

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

Hei kimi i te tauaro o x-y, kimihia te tauaro o ia taurangi.

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

Pahekotia te 2x me -x, ka x.

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

Pahekotia te 2y me y, ka 3y.

$$x+3y=3$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te x-y.

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

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

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

Pahekotia te y me 2y, ka 3y.

$$-x+3y=1$$

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.

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

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

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

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.

$$x+x=3-1$$

Tāpiri x ki te x.

$$2x=3-1$$

Tāpiri 3 ki te -1.

$$2x=2$$

Whakawehea ngā taha e rua ki te 2.

$$x=1$$

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

$$-1+3y=1$$

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

$$3y=2$$

Whakawehea ngā taha e rua ki te 3.

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

Kua oti te pūnaha te whakatau.

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