Whakaoti i te {l}{9x-4y=8}{6x-2y=3}

$$\left\{ \begin{array} { l } { 9 x - 4 y = 8 } \\ { 6 x - 2 y = 3 } \end{array} \right.$$

Whakaoti mō x, y

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

$$y = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

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.

$$9x-4y=8,6x-2y=3$$

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

$$9x-4y=8$$

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

$$9x=4y+8$$

Whakawehea ngā taha e rua ki te 9.

$$x=\frac{1}{9}\left(4y+8\right)$$

Whakareatia \frac{1}{9} ki te 8+4y.

$$x=\frac{4}{9}y+\frac{8}{9}$$

Whakakapia te \frac{8+4y}{9} mō te x ki tērā atu whārite, 6x-2y=3.

$$6\left(\frac{4}{9}y+\frac{8}{9}\right)-2y=3$$

Whakareatia 6 ki te \frac{8+4y}{9}.

$$\frac{8}{3}y+\frac{16}{3}-2y=3$$

Tāpiri \frac{8y}{3} ki te -2y.

$$\frac{2}{3}y+\frac{16}{3}=3$$

Me tango \frac{16}{3} mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua o te whārite ki te \frac{2}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{7}{2}$$

Whakaurua te -\frac{7}{2} mō y ki x=\frac{4}{9}y+\frac{8}{9}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{4}{9}\left(-\frac{7}{2}\right)+\frac{8}{9}$$

Whakareatia \frac{4}{9} ki te -\frac{7}{2} 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.

$$x=\frac{-14+8}{9}$$

Tāpiri \frac{8}{9} ki te -\frac{14}{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.

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

Kua oti te pūnaha te whakatau.

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

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.

$$9x-4y=8,6x-2y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}9&-4\\6&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}9&-4\\6&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&-4\\6&-2\end{matrix}\right))\left(\begin{matrix}9&-4\\6&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-4\\6&-2\end{matrix}\right))\left(\begin{matrix}8\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-4\\6&-2\end{matrix}\right))\left(\begin{matrix}8\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-4\\6&-2\end{matrix}\right))\left(\begin{matrix}8\\3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{9\left(-2\right)-\left(-4\times 6\right)}&-\frac{-4}{9\left(-2\right)-\left(-4\times 6\right)}\\-\frac{6}{9\left(-2\right)-\left(-4\times 6\right)}&\frac{9}{9\left(-2\right)-\left(-4\times 6\right)}\end{matrix}\right)\left(\begin{matrix}8\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&\frac{2}{3}\\-1&\frac{3}{2}\end{matrix}\right)\left(\begin{matrix}8\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}\times 8+\frac{2}{3}\times 3\\-8+\frac{3}{2}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$9x-4y=8,6x-2y=3$$

Kia ōrite ai a 9x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 9.

$$6\times 9x+6\left(-4\right)y=6\times 8,9\times 6x+9\left(-2\right)y=9\times 3$$

Whakarūnātia.

$$54x-24y=48,54x-18y=27$$

Me tango 54x-18y=27 mai i 54x-24y=48 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$54x-54x-24y+18y=48-27$$

Tāpiri 54x ki te -54x. Ka whakakore atu ngā kupu 54x me -54x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-24y+18y=48-27$$

Tāpiri -24y ki te 18y.

$$-6y=48-27$$

Tāpiri 48 ki te -27.

$$-6y=21$$

Whakawehea ngā taha e rua ki te -6.

$$y=-\frac{7}{2}$$

Whakaurua te -\frac{7}{2} mō y ki 6x-2y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x-2\left(-\frac{7}{2}\right)=3$$

Whakareatia -2 ki te -\frac{7}{2}.

$$6x+7=3$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$6x=-4$$

Whakawehea ngā taha e rua ki te 6.

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

Kua oti te pūnaha te whakatau.

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