Whakaoti i te {l}{3x+4y=-10}{x-4y=2}

$$\left. \begin{array} { l } { 3 x + 4 y = - 10 } \\ { x - 4 y = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-2$$

$$y=-1$$

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.

$$3x+4y=-10,x-4y=2$$

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

$$3x+4y=-10$$

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

$$3x=-4y-10$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-4y-10\right)$$

Whakareatia \frac{1}{3} ki te -4y-10.

$$x=-\frac{4}{3}y-\frac{10}{3}$$

Whakakapia te \frac{-4y-10}{3} mō te x ki tērā atu whārite, x-4y=2.

$$-\frac{4}{3}y-\frac{10}{3}-4y=2$$

Tāpiri -\frac{4y}{3} ki te -4y.

$$-\frac{16}{3}y-\frac{10}{3}=2$$

Me tāpiri \frac{10}{3} ki ngā taha e rua o te whārite.

$$-\frac{16}{3}y=\frac{16}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{16}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-1$$

Whakaurua te -1 mō y ki x=-\frac{4}{3}y-\frac{10}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{4}{3}\left(-1\right)-\frac{10}{3}$$

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

$$x=\frac{4-10}{3}$$

Tāpiri -\frac{10}{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.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=-1$$

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.

$$3x+4y=-10,x-4y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&4\\1&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-10\\2\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&4\\1&-4\end{matrix}\right))\left(\begin{matrix}3&4\\1&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\1&-4\end{matrix}\right))\left(\begin{matrix}-10\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\1&-4\end{matrix}\right))\left(\begin{matrix}-10\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\1&-4\end{matrix}\right))\left(\begin{matrix}-10\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{3\left(-4\right)-4}&-\frac{4}{3\left(-4\right)-4}\\-\frac{1}{3\left(-4\right)-4}&\frac{3}{3\left(-4\right)-4}\end{matrix}\right)\left(\begin{matrix}-10\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&\frac{1}{4}\\\frac{1}{16}&-\frac{3}{16}\end{matrix}\right)\left(\begin{matrix}-10\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\left(-10\right)+\frac{1}{4}\times 2\\\frac{1}{16}\left(-10\right)-\frac{3}{16}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-2,y=-1$$

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.

$$3x+4y=-10,x-4y=2$$

Kia ōrite ai a 3x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$3x+4y=-10,3x+3\left(-4\right)y=3\times 2$$

Whakarūnātia.

$$3x+4y=-10,3x-12y=6$$

Me tango 3x-12y=6 mai i 3x+4y=-10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+4y+12y=-10-6$$

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.

$$4y+12y=-10-6$$

Tāpiri 4y ki te 12y.

$$16y=-10-6$$

Tāpiri -10 ki te -6.

$$16y=-16$$

Whakawehea ngā taha e rua ki te 16.

$$y=-1$$

Whakaurua te -1 mō y ki x-4y=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-4\left(-1\right)=2$$

Whakareatia -4 ki te -1.

$$x+4=2$$

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

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=-1$$