Whakaoti i te {l}{x+y=7}{3x+y=17}

$$\left\{ \begin{array} { l } { x + y = 7 } \\ { 3 x + y = 17 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=2$$

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.

$$x+y=7,3x+y=17$$

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

$$x+y=7$$

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

$$x=-y+7$$

Whakakapia te -y+7 mō te x ki tērā atu whārite, 3x+y=17.

$$3\left(-y+7\right)+y=17$$

Whakareatia 3 ki te -y+7.

$$-3y+21+y=17$$

Tāpiri -3y ki te y.

$$-2y+21=17$$

Me tango 21 mai i ngā taha e rua o te whārite.

$$-2y=-4$$

Whakawehea ngā taha e rua ki te -2.

$$y=2$$

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

$$x=-2+7$$

Tāpiri 7 ki te -2.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=2$$

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.

$$x+y=7,3x+y=17$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\17\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\3&1\end{matrix}\right))\left(\begin{matrix}1&1\\3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\3&1\end{matrix}\right))\left(\begin{matrix}7\\17\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\3&1\end{matrix}\right))\left(\begin{matrix}7\\17\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\3&1\end{matrix}\right))\left(\begin{matrix}7\\17\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-3}&-\frac{1}{1-3}\\-\frac{3}{1-3}&\frac{1}{1-3}\end{matrix}\right)\left(\begin{matrix}7\\17\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}&\frac{1}{2}\\\frac{3}{2}&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}7\\17\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}\times 7+\frac{1}{2}\times 17\\\frac{3}{2}\times 7-\frac{1}{2}\times 17\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=5,y=2$$

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.

$$x+y=7,3x+y=17$$

Me tango 3x+y=17 mai i x+y=7 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-3x+y-y=7-17$$

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

$$x-3x=7-17$$

Tāpiri x ki te -3x.

$$-2x=7-17$$

Tāpiri 7 ki te -17.

$$-2x=-10$$

Whakawehea ngā taha e rua ki te -2.

$$x=5$$

Whakaurua te 5 mō x ki 3x+y=17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$3\times 5+y=17$$

Whakareatia 3 ki te 5.

$$15+y=17$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$y=2$$

Kua oti te pūnaha te whakatau.

$$x=5,y=2$$