Whakaoti i te {l}{3x+y=4}{6x+y=4}

$$\left. \begin{array} { l } { 3 x + y = 4 } \\ { 6 x + y = 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=4$$

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.

$$3x+y=4,6x+y=4$$

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

$$3x+y=4$$

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

$$3x=-y+4$$

Whakawehea ngā taha e rua ki te 3.

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

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

$$x=-\frac{1}{3}y+\frac{4}{3}$$

Whakakapia te \frac{-y+4}{3} mō te x ki tērā atu whārite, 6x+y=4.

$$6\left(-\frac{1}{3}y+\frac{4}{3}\right)+y=4$$

Whakareatia 6 ki te \frac{-y+4}{3}.

$$-2y+8+y=4$$

Tāpiri -2y ki te y.

$$-y+8=4$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$-y=-4$$

Whakawehea ngā taha e rua ki te -1.

$$y=4$$

Whakaurua te 4 mō y ki x=-\frac{1}{3}y+\frac{4}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{3}\times 4+\frac{4}{3}$$

Whakareatia -\frac{1}{3} ki te 4.

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

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

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=4$$

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.

$$3x+y=4,6x+y=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&1\\6&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&1\\6&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&1\\6&1\end{matrix}\right))\left(\begin{matrix}3&1\\6&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\6&1\end{matrix}\right))\left(\begin{matrix}4\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\6&1\end{matrix}\right))\left(\begin{matrix}4\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\6&1\end{matrix}\right))\left(\begin{matrix}4\\4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3-6}&-\frac{1}{3-6}\\-\frac{6}{3-6}&\frac{3}{3-6}\end{matrix}\right)\left(\begin{matrix}4\\4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&\frac{1}{3}\\2&-1\end{matrix}\right)\left(\begin{matrix}4\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}\times 4+\frac{1}{3}\times 4\\2\times 4-4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=0,y=4$$

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.

$$3x+y=4,6x+y=4$$

Me tango 6x+y=4 mai i 3x+y=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-6x+y-y=4-4$$

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

$$3x-6x=4-4$$

Tāpiri 3x ki te -6x.

$$-3x=4-4$$

Tāpiri 4 ki te -4.

$$-3x=0$$

Whakawehea ngā taha e rua ki te -3.

$$x=0$$

Whakaurua te 0 mō x ki 6x+y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=4$$

Kua oti te pūnaha te whakatau.

$$x=0,y=4$$