Whakaoti i te {l}{x+2y=0}{5x+2y=3}

$$\left\{ \begin{array} { l } { x + 2 y = 0 } \\ { 5 x + 2 y = 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{3}{4}=0.75$$

$$y=-\frac{3}{8}=-0.375$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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.

$$x+2y=0,5x+2y=3$$

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

$$x+2y=0$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$x=-2y$$

Whakakapia te -2y mō te x ki tērā atu whārite, 5x+2y=3.

$$5\left(-2\right)y+2y=3$$

Whakareatia 5 ki te -2y.

$$-10y+2y=3$$

Tāpiri -10y ki te 2y.

$$-8y=3$$

Whakawehea ngā taha e rua ki te -8.

$$y=-\frac{3}{8}$$

Whakaurua te -\frac{3}{8} mō y ki x=-2y. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\left(-\frac{3}{8}\right)$$

Whakareatia -2 ki te -\frac{3}{8}.

$$x=\frac{3}{4}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3}{4},y=-\frac{3}{8}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$x+2y=0,5x+2y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&2\\5&2\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&\frac{1}{4}\\\frac{5}{8}&-\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}0\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 3\\-\frac{1}{8}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}\\-\frac{3}{8}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{3}{4},y=-\frac{3}{8}$$

Ngā Upane e Whakamahi i te Tangohanga

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.

$$x+2y=0,5x+2y=3$$

Me tango 5x+2y=3 mai i x+2y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

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

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

$$x-5x=-3$$

Tāpiri x ki te -5x.

$$-4x=-3$$

Whakawehea ngā taha e rua ki te -4.

$$x=\frac{3}{4}$$

Whakaurua te \frac{3}{4} mō x ki 5x+2y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$5\times \frac{3}{4}+2y=3$$

Whakareatia 5 ki te \frac{3}{4}.

$$\frac{15}{4}+2y=3$$

Me tango \frac{15}{4} mai i ngā taha e rua o te whārite.

$$2y=-\frac{3}{4}$$

Whakawehea ngā taha e rua ki te 2.

$$y=-\frac{3}{8}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3}{4},y=-\frac{3}{8}$$