Whakaoti i te {l}{-3x+y=1}{-3x+2y=5}

$$\left. \begin{array} { l } { - 3 x + y = 1 } \\ { - 3 x + 2 y = 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=4$$

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.

$$-3x+y=1,-3x+2y=5$$

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

$$-3x+y=1$$

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

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

Whakawehea ngā taha e rua ki te -3.

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

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

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

Whakakapia te \frac{-1+y}{3} mō te x ki tērā atu whārite, -3x+2y=5.

$$-3\left(\frac{1}{3}y-\frac{1}{3}\right)+2y=5$$

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

$$-y+1+2y=5$$

Tāpiri -y ki te 2y.

$$y+1=5$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$y=4$$

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

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

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

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

Tāpiri -\frac{1}{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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=4$$

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.

$$-3x+y=1,-3x+2y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-3&1\\-3&2\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=4$$

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.

$$-3x+y=1,-3x+2y=5$$

Me tango -3x+2y=5 mai i -3x+y=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-3x+3x+y-2y=1-5$$

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.

$$y-2y=1-5$$

Tāpiri y ki te -2y.

$$-y=1-5$$

Tāpiri 1 ki te -5.

$$-y=-4$$

Whakawehea ngā taha e rua ki te -1.

$$y=4$$

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

$$-3x+2\times 4=5$$

Whakareatia 2 ki te 4.

$$-3x+8=5$$

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

$$-3x=-3$$

Whakawehea ngā taha e rua ki te -3.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=4$$