Whakaoti i te {l}{4x+y=4}{-3x-6y=18}

$$\left. \begin{array} { l } { 4 x + y = 4 } \\ { - 3 x - 6 y = 18 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=-4$$

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.

$$4x+y=4,-3x-6y=18$$

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 tango y mai i 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, -3x-6y=18.

$$-3\left(-\frac{1}{4}y+1\right)-6y=18$$

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

$$\frac{3}{4}y-3-6y=18$$

Tāpiri \frac{3y}{4} ki te -6y.

$$-\frac{21}{4}y-3=18$$

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

$$-\frac{21}{4}y=21$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{21}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-4$$

Whakaurua te -4 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.

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

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

$$x=1+1$$

Tāpiri 1 ki te 1.

$$x=2$$

Kua oti te pūnaha te whakatau.

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

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.

$$4x+y=4,-3x-6y=18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&1\\-3&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&1\\-3&-6\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&1\\-3&-6\end{matrix}\right))\left(\begin{matrix}4&1\\-3&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&1\\-3&-6\end{matrix}\right))\left(\begin{matrix}4\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&1\\-3&-6\end{matrix}\right))\left(\begin{matrix}4\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&1\\-3&-6\end{matrix}\right))\left(\begin{matrix}4\\18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{4\left(-6\right)-\left(-3\right)}&-\frac{1}{4\left(-6\right)-\left(-3\right)}\\-\frac{-3}{4\left(-6\right)-\left(-3\right)}&\frac{4}{4\left(-6\right)-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}4\\18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}&\frac{1}{21}\\-\frac{1}{7}&-\frac{4}{21}\end{matrix}\right)\left(\begin{matrix}4\\18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}\times 4+\frac{1}{21}\times 18\\-\frac{1}{7}\times 4-\frac{4}{21}\times 18\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$4x+y=4,-3x-6y=18$$

Kia ōrite ai a 4x me -3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$-3\times 4x-3y=-3\times 4,4\left(-3\right)x+4\left(-6\right)y=4\times 18$$

Whakarūnātia.

$$-12x-3y=-12,-12x-24y=72$$

Me tango -12x-24y=72 mai i -12x-3y=-12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-12x+12x-3y+24y=-12-72$$

Tāpiri -12x ki te 12x. Ka whakakore atu ngā kupu -12x me 12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-3y+24y=-12-72$$

Tāpiri -3y ki te 24y.

$$21y=-12-72$$

Tāpiri -12 ki te -72.

$$21y=-84$$

Whakawehea ngā taha e rua ki te 21.

$$y=-4$$

Whakaurua te -4 mō y ki -3x-6y=18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-3x-6\left(-4\right)=18$$

Whakareatia -6 ki te -4.

$$-3x+24=18$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$-3x=-6$$

Whakawehea ngā taha e rua ki te -3.

$$x=2$$

Kua oti te pūnaha te whakatau.

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