Whakaoti i te {l}{x-4y=4}{7x-7y=-14}

$$\left. \begin{array} { l } { x - 4 y = 4 } \\ { 7 x - 7 y = - 14 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-4$$

$$y=-2$$

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.

$$x-4y=4,7x-7y=-14$$

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

$$x-4y=4$$

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

$$x=4y+4$$

Whakakapia te 4+4y mō te x ki tērā atu whārite, 7x-7y=-14.

$$7\left(4y+4\right)-7y=-14$$

Whakareatia 7 ki te 4+4y.

$$28y+28-7y=-14$$

Tāpiri 28y ki te -7y.

$$21y+28=-14$$

Me tango 28 mai i ngā taha e rua o te whārite.

$$21y=-42$$

Whakawehea ngā taha e rua ki te 21.

$$y=-2$$

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

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

Whakareatia 4 ki te -2.

$$x=-8+4$$

Tāpiri 4 ki te -8.

$$x=-4$$

Kua oti te pūnaha te whakatau.

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

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.

$$x-4y=4,7x-7y=-14$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-4\\7&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\-14\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-4\\7&-7\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-4\\7&-7\end{matrix}\right))\left(\begin{matrix}1&-4\\7&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\7&-7\end{matrix}\right))\left(\begin{matrix}4\\-14\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\7&-7\end{matrix}\right))\left(\begin{matrix}4\\-14\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-4\\7&-7\end{matrix}\right))\left(\begin{matrix}4\\-14\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{-7-\left(-4\times 7\right)}&-\frac{-4}{-7-\left(-4\times 7\right)}\\-\frac{7}{-7-\left(-4\times 7\right)}&\frac{1}{-7-\left(-4\times 7\right)}\end{matrix}\right)\left(\begin{matrix}4\\-14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&\frac{4}{21}\\-\frac{1}{3}&\frac{1}{21}\end{matrix}\right)\left(\begin{matrix}4\\-14\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}\times 4+\frac{4}{21}\left(-14\right)\\-\frac{1}{3}\times 4+\frac{1}{21}\left(-14\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$x-4y=4,7x-7y=-14$$

Kia ōrite ai a x me 7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$7x+7\left(-4\right)y=7\times 4,7x-7y=-14$$

Whakarūnātia.

$$7x-28y=28,7x-7y=-14$$

Me tango 7x-7y=-14 mai i 7x-28y=28 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$7x-7x-28y+7y=28+14$$

Tāpiri 7x ki te -7x. Ka whakakore atu ngā kupu 7x me -7x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-28y+7y=28+14$$

Tāpiri -28y ki te 7y.

$$-21y=28+14$$

Tāpiri 28 ki te 14.

$$-21y=42$$

Whakawehea ngā taha e rua ki te -21.

$$y=-2$$

Whakaurua te -2 mō y ki 7x-7y=-14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x-7\left(-2\right)=-14$$

Whakareatia -7 ki te -2.

$$7x+14=-14$$

Me tango 14 mai i ngā taha e rua o te whārite.

$$7x=-28$$

Whakawehea ngā taha e rua ki te 7.

$$x=-4$$

Kua oti te pūnaha te whakatau.

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