Whakaoti i te {c}{4x-y=4}{-12x+2y=-3}

$$\left. \begin{array} { c } { 4 x - y = 4 } \\ { - 12 x + 2 y = - 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{5}{4} = -1\frac{1}{4} = -1.25$$

$$y=-9$$

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.

$$4x-y=4,-12x+2y=-3$$

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

$$4x-y=4$$

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

$$4x=y+4$$

Whakawehea ngā taha e rua ki te 4.

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

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

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

Whakakapia te \frac{y}{4}+1 mō te x ki tērā atu whārite, -12x+2y=-3.

$$-12\left(\frac{1}{4}y+1\right)+2y=-3$$

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

$$-3y-12+2y=-3$$

Tāpiri -3y ki te 2y.

$$-y-12=-3$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$-y=9$$

Whakawehea ngā taha e rua ki te -1.

$$y=-9$$

Whakaurua te -9 mō y ki x=\frac{1}{4}y+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{4}\left(-9\right)+1$$

Whakareatia \frac{1}{4} ki te -9.

$$x=-\frac{9}{4}+1$$

Tāpiri 1 ki te -\frac{9}{4}.

$$x=-\frac{5}{4}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{5}{4},y=-9$$

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.

$$4x-y=4,-12x+2y=-3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-1\\-12&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\-3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-1\\-12&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&-1\\-12&2\end{matrix}\right))\left(\begin{matrix}4&-1\\-12&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\-12&2\end{matrix}\right))\left(\begin{matrix}4\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\-12&2\end{matrix}\right))\left(\begin{matrix}4\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\-12&2\end{matrix}\right))\left(\begin{matrix}4\\-3\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te \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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{4\times 2-\left(-\left(-12\right)\right)}&-\frac{-1}{4\times 2-\left(-\left(-12\right)\right)}\\-\frac{-12}{4\times 2-\left(-\left(-12\right)\right)}&\frac{4}{4\times 2-\left(-\left(-12\right)\right)}\end{matrix}\right)\left(\begin{matrix}4\\-3\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{4}\\-9\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{5}{4},y=-9$$

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.

$$4x-y=4,-12x+2y=-3$$

Kia ōrite ai a 4x me -12x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -12 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$-12\times 4x-12\left(-1\right)y=-12\times 4,4\left(-12\right)x+4\times 2y=4\left(-3\right)$$

Whakarūnātia.

$$-48x+12y=-48,-48x+8y=-12$$

Me tango -48x+8y=-12 mai i -48x+12y=-48 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-48x+48x+12y-8y=-48+12$$

Tāpiri -48x ki te 48x. Ka whakakore atu ngā kupu -48x me 48x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$12y-8y=-48+12$$

Tāpiri 12y ki te -8y.

$$4y=-48+12$$

Tāpiri -48 ki te 12.

$$4y=-36$$

Whakawehea ngā taha e rua ki te 4.

$$y=-9$$

Whakaurua te -9 mō y ki -12x+2y=-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-12x+2\left(-9\right)=-3$$

Whakareatia 2 ki te -9.

$$-12x-18=-3$$

Me tāpiri 18 ki ngā taha e rua o te whārite.

$$-12x=15$$

Whakawehea ngā taha e rua ki te -12.

$$x=-\frac{5}{4}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{5}{4},y=-9$$