Whakaoti i te {l}{3x-4y=12}{y=-3x-3}

$$\left. \begin{array} { l } { 3 x - 4 y = 12 } \\ { y = - 3 x - 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=-3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Me tāpiri te 3x ki ngā taha e rua.

$$y+3x=-3$$

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.

$$3x-4y=12,3x+y=-3$$

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

$$3x-4y=12$$

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

$$3x=4y+12$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(4y+12\right)$$

Whakareatia \frac{1}{3} ki te 12+4y.

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

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

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

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

$$4y+12+y=-3$$

Tāpiri 4y ki te y.

$$5y+12=-3$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$5y=-15$$

Whakawehea ngā taha e rua ki te 5.

$$y=-3$$

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

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

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

$$x=-4+4$$

Tāpiri 4 ki te -4.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-3$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Me tāpiri te 3x ki ngā taha e rua.

$$y+3x=-3$$

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

$$3x-4y=12,3x+y=-3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-4\\3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\-3\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&-4\\3&1\end{matrix}\right))\left(\begin{matrix}3&-4\\3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-4\\3&1\end{matrix}\right))\left(\begin{matrix}12\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-4\\3&1\end{matrix}\right))\left(\begin{matrix}12\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-4\\3&1\end{matrix}\right))\left(\begin{matrix}12\\-3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3-\left(-4\times 3\right)}&-\frac{-4}{3-\left(-4\times 3\right)}\\-\frac{3}{3-\left(-4\times 3\right)}&\frac{3}{3-\left(-4\times 3\right)}\end{matrix}\right)\left(\begin{matrix}12\\-3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{15}&\frac{4}{15}\\-\frac{1}{5}&\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}12\\-3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{15}\times 12+\frac{4}{15}\left(-3\right)\\-\frac{1}{5}\times 12+\frac{1}{5}\left(-3\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=0,y=-3$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Me tāpiri te 3x ki ngā taha e rua.

$$y+3x=-3$$

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.

$$3x-4y=12,3x+y=-3$$

Me tango 3x+y=-3 mai i 3x-4y=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x-4y-y=12+3$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-4y-y=12+3$$

Tāpiri -4y ki te -y.

$$-5y=12+3$$

Tāpiri 12 ki te 3.

$$-5y=15$$

Whakawehea ngā taha e rua ki te -5.

$$y=-3$$

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

$$3x-3=-3$$

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

$$3x=0$$

Whakawehea ngā taha e rua ki te 3.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-3$$