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

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

Whakaoti mō x, y

$$x = \frac{11}{2} = 5\frac{1}{2} = 5.5$$

$$y=\frac{1}{2}=0.5$$

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.

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

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

$$x+3y=7$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$x=-3y+7$$

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

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

Whakareatia 3 ki te -3y+7.

$$-9y+21+y=17$$

Tāpiri -9y ki te y.

$$-8y+21=17$$

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

$$-8y=-4$$

Whakawehea ngā taha e rua ki te -8.

$$y=\frac{1}{2}$$

Whakaurua te \frac{1}{2} mō y ki x=-3y+7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-3\times \frac{1}{2}+7$$

Whakareatia -3 ki te \frac{1}{2}.

$$x=-\frac{3}{2}+7$$

Tāpiri 7 ki te -\frac{3}{2}.

$$x=\frac{11}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{11}{2},y=\frac{1}{2}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{8}&\frac{3}{8}\\\frac{3}{8}&-\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}7\\17\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{8}\times 7+\frac{3}{8}\times 17\\\frac{3}{8}\times 7-\frac{1}{8}\times 17\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{11}{2}\\\frac{1}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{11}{2},y=\frac{1}{2}$$

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.

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

Kia ōrite ai a x 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 1.

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

Whakarūnātia.

$$3x+9y=21,3x+y=17$$

Me tango 3x+y=17 mai i 3x+9y=21 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+9y-y=21-17$$

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.

$$9y-y=21-17$$

Tāpiri 9y ki te -y.

$$8y=21-17$$

Tāpiri 21 ki te -17.

$$8y=4$$

Whakawehea ngā taha e rua ki te 8.

$$y=\frac{1}{2}$$

Whakaurua te \frac{1}{2} mō y ki 3x+y=17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+\frac{1}{2}=17$$

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$3x=\frac{33}{2}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{11}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{11}{2},y=\frac{1}{2}$$