Whakaoti i te {l}{x-4y=5}{-2x-y=-4}

$$\left\{ \begin{array} { l } { x - 4 y = 5 } \\ { - 2 x - y = - 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{7}{3} = 2\frac{1}{3} \approx 2.333333333$$

$$y=-\frac{2}{3}\approx -0.666666667$$

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.

$$x-4y=5,-2x-y=-4$$

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

$$x-4y=5$$

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

$$x=4y+5$$

Whakakapia te 4y+5 mō te x ki tērā atu whārite, -2x-y=-4.

$$-2\left(4y+5\right)-y=-4$$

Whakareatia -2 ki te 4y+5.

$$-8y-10-y=-4$$

Tāpiri -8y ki te -y.

$$-9y-10=-4$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$-9y=6$$

Whakawehea ngā taha e rua ki te -9.

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

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

$$x=4\left(-\frac{2}{3}\right)+5$$

Whakareatia 4 ki te -\frac{2}{3}.

$$x=-\frac{8}{3}+5$$

Tāpiri 5 ki te -\frac{8}{3}.

$$x=\frac{7}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{7}{3},y=-\frac{2}{3}$$

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.

$$x-4y=5,-2x-y=-4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-4\\-2&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\-4\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-4\\-2&-1\end{matrix}\right))\left(\begin{matrix}1&-4\\-2&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\-2&-1\end{matrix}\right))\left(\begin{matrix}5\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\-2&-1\end{matrix}\right))\left(\begin{matrix}5\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\-2&-1\end{matrix}\right))\left(\begin{matrix}5\\-4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-\left(-4\left(-2\right)\right)}&-\frac{-4}{-1-\left(-4\left(-2\right)\right)}\\-\frac{-2}{-1-\left(-4\left(-2\right)\right)}&\frac{1}{-1-\left(-4\left(-2\right)\right)}\end{matrix}\right)\left(\begin{matrix}5\\-4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}&-\frac{4}{9}\\-\frac{2}{9}&-\frac{1}{9}\end{matrix}\right)\left(\begin{matrix}5\\-4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}\times 5-\frac{4}{9}\left(-4\right)\\-\frac{2}{9}\times 5-\frac{1}{9}\left(-4\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{3}\\-\frac{2}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{7}{3},y=-\frac{2}{3}$$

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.

$$x-4y=5,-2x-y=-4$$

Kia ōrite ai a x me -2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$-2x-2\left(-4\right)y=-2\times 5,-2x-y=-4$$

Whakarūnātia.

$$-2x+8y=-10,-2x-y=-4$$

Me tango -2x-y=-4 mai i -2x+8y=-10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x+8y+y=-10+4$$

Tāpiri -2x ki te 2x. Ka whakakore atu ngā kupu -2x me 2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$8y+y=-10+4$$

Tāpiri 8y ki te y.

$$9y=-10+4$$

Tāpiri -10 ki te 4.

$$9y=-6$$

Whakawehea ngā taha e rua ki te 9.

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

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

$$-2x-\left(-\frac{2}{3}\right)=-4$$

Me tango \frac{2}{3} mai i ngā taha e rua o te whārite.

$$-2x=-\frac{14}{3}$$

Whakawehea ngā taha e rua ki te -2.

$$x=\frac{7}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{7}{3},y=-\frac{2}{3}$$