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

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

Whakaoti mō x, y

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

$$y=-2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 4x mai i ngā taha e rua.

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

Pahekotia te 2x me -4x, ka -2x.

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

Whakaarohia te whārite tuarua. Whakawehea ngā taha e rua ki te 2.

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

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.

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

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

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

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

$$-2x=y-3$$

Whakawehea ngā taha e rua ki te -2.

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

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

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

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

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

Tāpiri -\frac{y}{2} ki te y.

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

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

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

Me whakarea ngā taha e rua ki te 2.

$$y=-2$$

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

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

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

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

Tāpiri \frac{3}{2} ki te 1.

$$x=\frac{5}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{2},y=-2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 4x mai i ngā taha e rua.

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

Pahekotia te 2x me -4x, ka -2x.

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

Whakaarohia te whārite tuarua. Whakawehea ngā taha e rua ki te 2.

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

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=\frac{5}{2},y=-2$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 4x mai i ngā taha e rua.

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

Pahekotia te 2x me -4x, ka -2x.

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

Whakaarohia te whārite tuarua. Whakawehea ngā taha e rua ki te 2.

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

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.

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

Kia ōrite ai a -2x me x, 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.

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

Whakarūnātia.

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

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

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

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

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

Tāpiri -y ki te 2y.

$$y=-3+1$$

Tāpiri -3 ki te 1.

$$y=-2$$

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

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

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

$$x=\frac{5}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{2},y=-2$$