Whakaoti i te {l}{9x-3y-13=0}{2x+y-4=0}

$$\left. \begin{array} { l } { 9 x - 3 y - 13 = 0 } \\ { 2 x + y - 4 = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

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

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.

$$9x-3y-13=0,2x+y-4=0$$

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

$$9x-3y-13=0$$

Me tāpiri 13 ki ngā taha e rua o te whārite.

$$9x-3y=13$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$9x=3y+13$$

Whakawehea ngā taha e rua ki te 9.

$$x=\frac{1}{9}\left(3y+13\right)$$

Whakareatia \frac{1}{9} ki te 3y+13.

$$x=\frac{1}{3}y+\frac{13}{9}$$

Whakakapia te \frac{y}{3}+\frac{13}{9} mō te x ki tērā atu whārite, 2x+y-4=0.

$$2\left(\frac{1}{3}y+\frac{13}{9}\right)+y-4=0$$

Whakareatia 2 ki te \frac{y}{3}+\frac{13}{9}.

$$\frac{2}{3}y+\frac{26}{9}+y-4=0$$

Tāpiri \frac{2y}{3} ki te y.

$$\frac{5}{3}y+\frac{26}{9}-4=0$$

Tāpiri \frac{26}{9} ki te -4.

$$\frac{5}{3}y-\frac{10}{9}=0$$

Me tāpiri \frac{10}{9} ki ngā taha e rua o te whārite.

$$\frac{5}{3}y=\frac{10}{9}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

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

Whakaurua te \frac{2}{3} mō y ki x=\frac{1}{3}y+\frac{13}{9}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{3}\times \frac{2}{3}+\frac{13}{9}$$

Whakareatia \frac{1}{3} ki te \frac{2}{3} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{2+13}{9}$$

Tāpiri \frac{13}{9} ki te \frac{2}{9} 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.

$$x=\frac{5}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{3},y=\frac{2}{3}$$

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.

$$9x-3y-13=0,2x+y-4=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}9&-3\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}13\\4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}9&-3\\2&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&-3\\2&1\end{matrix}\right))\left(\begin{matrix}9&-3\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-3\\2&1\end{matrix}\right))\left(\begin{matrix}13\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-3\\2&1\end{matrix}\right))\left(\begin{matrix}13\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-3\\2&1\end{matrix}\right))\left(\begin{matrix}13\\4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9-\left(-3\times 2\right)}&-\frac{-3}{9-\left(-3\times 2\right)}\\-\frac{2}{9-\left(-3\times 2\right)}&\frac{9}{9-\left(-3\times 2\right)}\end{matrix}\right)\left(\begin{matrix}13\\4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{15}&\frac{1}{5}\\-\frac{2}{15}&\frac{3}{5}\end{matrix}\right)\left(\begin{matrix}13\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{15}\times 13+\frac{1}{5}\times 4\\-\frac{2}{15}\times 13+\frac{3}{5}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{3}\\\frac{2}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{5}{3},y=\frac{2}{3}$$

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.

$$9x-3y-13=0,2x+y-4=0$$

Kia ōrite ai a 9x 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 9.

$$2\times 9x+2\left(-3\right)y+2\left(-13\right)=0,9\times 2x+9y+9\left(-4\right)=0$$

Whakarūnātia.

$$18x-6y-26=0,18x+9y-36=0$$

Me tango 18x+9y-36=0 mai i 18x-6y-26=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$18x-18x-6y-9y-26+36=0$$

Tāpiri 18x ki te -18x. Ka whakakore atu ngā kupu 18x me -18x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-6y-9y-26+36=0$$

Tāpiri -6y ki te -9y.

$$-15y-26+36=0$$

Tāpiri -26 ki te 36.

$$-15y+10=0$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$-15y=-10$$

Whakawehea ngā taha e rua ki te -15.

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

Whakaurua te \frac{2}{3} mō y ki 2x+y-4=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+\frac{2}{3}-4=0$$

Tāpiri \frac{2}{3} ki te -4.

$$2x-\frac{10}{3}=0$$

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

$$2x=\frac{10}{3}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{5}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{3},y=\frac{2}{3}$$