Whakaoti i te {l}{4x-3y=21}{x=-y}

$$\left. \begin{array} { l } { 4 x - 3 y = 21 } \\ { x = - y } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=-3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Me tāpiri te y ki ngā taha e rua.

$$x+y=0$$

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.

$$4x-3y=21,x+y=0$$

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

$$4x-3y=21$$

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

$$4x=3y+21$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(3y+21\right)$$

Whakareatia \frac{1}{4} ki te 21+3y.

$$x=\frac{3}{4}y+\frac{21}{4}$$

Whakakapia te \frac{21+3y}{4} mō te x ki tērā atu whārite, x+y=0.

$$\frac{3}{4}y+\frac{21}{4}+y=0$$

Tāpiri \frac{3y}{4} ki te y.

$$\frac{7}{4}y+\frac{21}{4}=0$$

Me tango \frac{21}{4} mai i ngā taha e rua o te whārite.

$$\frac{7}{4}y=-\frac{21}{4}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-3$$

Whakaurua te -3 mō y ki x=\frac{3}{4}y+\frac{21}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{4}\left(-3\right)+\frac{21}{4}$$

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

$$x=\frac{-9+21}{4}$$

Tāpiri \frac{21}{4} ki te -\frac{9}{4} 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.

$$x=3$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Me tāpiri te y ki ngā taha e rua.

$$x+y=0$$

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

$$4x-3y=21,x+y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-3\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}21\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-3\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&-3\\1&1\end{matrix}\right))\left(\begin{matrix}4&-3\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-3\\1&1\end{matrix}\right))\left(\begin{matrix}21\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-3\\1&1\end{matrix}\right))\left(\begin{matrix}21\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-3\\1&1\end{matrix}\right))\left(\begin{matrix}21\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4-\left(-3\right)}&-\frac{-3}{4-\left(-3\right)}\\-\frac{1}{4-\left(-3\right)}&\frac{4}{4-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}21\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}&\frac{3}{7}\\-\frac{1}{7}&\frac{4}{7}\end{matrix}\right)\left(\begin{matrix}21\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}\times 21\\-\frac{1}{7}\times 21\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Me tāpiri te y ki ngā taha e rua.

$$x+y=0$$

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.

$$4x-3y=21,x+y=0$$

Kia ōrite ai a 4x 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 4.

$$4x-3y=21,4x+4y=0$$

Me tango 4x+4y=0 mai i 4x-3y=21 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-4x-3y-4y=21$$

Tāpiri 4x ki te -4x. Ka whakakore atu ngā kupu 4x me -4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-3y-4y=21$$

Tāpiri -3y ki te -4y.

$$-7y=21$$

Whakawehea ngā taha e rua ki te -7.

$$y=-3$$

Whakaurua te -3 mō y ki x+y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-3=0$$

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

$$x=3$$

Kua oti te pūnaha te whakatau.

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