Whakaoti i te {r}{4x-2y+4=0}{-4x+3y-3=0}

$$\left. \begin{array} { r } { 4 x - 2 y + 4 = 0 } \\ { - 4 x + 3 y - 3 = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

$$y=-1$$

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.

$$4x-2y+4=0,-4x+3y-3=0$$

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

$$4x-2y+4=0$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$4x-2y=-4$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$4x=2y-4$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(2y-4\right)$$

Whakareatia \frac{1}{4} ki te -4+2y.

$$x=\frac{1}{2}y-1$$

Whakakapia te \frac{y}{2}-1 mō te x ki tērā atu whārite, -4x+3y-3=0.

$$-4\left(\frac{1}{2}y-1\right)+3y-3=0$$

Whakareatia -4 ki te \frac{y}{2}-1.

$$-2y+4+3y-3=0$$

Tāpiri -2y ki te 3y.

$$y+4-3=0$$

Tāpiri 4 ki te -3.

$$y+1=0$$

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

$$y=-1$$

Whakaurua te -1 mō y ki x=\frac{1}{2}y-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{2}\left(-1\right)-1$$

Whakareatia \frac{1}{2} ki te -1.

$$x=-\frac{1}{2}-1$$

Tāpiri -1 ki te -\frac{1}{2}.

$$x=-\frac{3}{2}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{3}{2},y=-1$$

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.

$$4x-2y+4=0,-4x+3y-3=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-2\\-4&3\end{matrix}\right).

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-2\\-4&3\end{matrix}\right))\left(\begin{matrix}-4\\3\end{matrix}\right)$$

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

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

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{3}{2},y=-1$$

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.

$$4x-2y+4=0,-4x+3y-3=0$$

Kia ōrite ai a 4x me -4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$-4\times 4x-4\left(-2\right)y-4\times 4=0,4\left(-4\right)x+4\times 3y+4\left(-3\right)=0$$

Whakarūnātia.

$$-16x+8y-16=0,-16x+12y-12=0$$

Me tango -16x+12y-12=0 mai i -16x+8y-16=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-16x+16x+8y-12y-16+12=0$$

Tāpiri -16x ki te 16x. Ka whakakore atu ngā kupu -16x me 16x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$8y-12y-16+12=0$$

Tāpiri 8y ki te -12y.

$$-4y-16+12=0$$

Tāpiri -16 ki te 12.

$$-4y-4=0$$

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

$$-4y=4$$

Whakawehea ngā taha e rua ki te -4.

$$y=-1$$

Whakaurua te -1 mō y ki -4x+3y-3=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-4x+3\left(-1\right)-3=0$$

Whakareatia 3 ki te -1.

$$-4x-3-3=0$$

Tāpiri -3 ki te -3.

$$-4x-6=0$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$-4x=6$$

Whakawehea ngā taha e rua ki te -4.

$$x=-\frac{3}{2}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{3}{2},y=-1$$