Whakaoti i te {l}{9x+7y=6}{8x+3y=9}

$$\left. \begin{array} { l } { 9 x + 7 y = 6 } \\ { 8 x + 3 y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{45}{29} = 1\frac{16}{29} \approx 1.551724138$$

$$y = -\frac{33}{29} = -1\frac{4}{29} \approx -1.137931034$$

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.

$$9x+7y=6,8x+3y=9$$

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

$$9x+7y=6$$

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

$$9x=-7y+6$$

Whakawehea ngā taha e rua ki te 9.

$$x=\frac{1}{9}\left(-7y+6\right)$$

Whakareatia \frac{1}{9} ki te -7y+6.

$$x=-\frac{7}{9}y+\frac{2}{3}$$

Whakakapia te -\frac{7y}{9}+\frac{2}{3} mō te x ki tērā atu whārite, 8x+3y=9.

$$8\left(-\frac{7}{9}y+\frac{2}{3}\right)+3y=9$$

Whakareatia 8 ki te -\frac{7y}{9}+\frac{2}{3}.

$$-\frac{56}{9}y+\frac{16}{3}+3y=9$$

Tāpiri -\frac{56y}{9} ki te 3y.

$$-\frac{29}{9}y+\frac{16}{3}=9$$

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

$$-\frac{29}{9}y=\frac{11}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{29}{9}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{33}{29}$$

Whakaurua te -\frac{33}{29} mō y ki x=-\frac{7}{9}y+\frac{2}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{7}{9}\left(-\frac{33}{29}\right)+\frac{2}{3}$$

Whakareatia -\frac{7}{9} ki te -\frac{33}{29} 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.

$$x=\frac{77}{87}+\frac{2}{3}$$

Tāpiri \frac{2}{3} ki te \frac{77}{87} 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.

$$x=\frac{45}{29}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{45}{29},y=-\frac{33}{29}$$

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.

$$9x+7y=6,8x+3y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}9&7\\8&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}9&7\\8&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&7\\8&3\end{matrix}\right))\left(\begin{matrix}9&7\\8&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&7\\8&3\end{matrix}\right))\left(\begin{matrix}6\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&7\\8&3\end{matrix}\right))\left(\begin{matrix}6\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&7\\8&3\end{matrix}\right))\left(\begin{matrix}6\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{9\times 3-7\times 8}&-\frac{7}{9\times 3-7\times 8}\\-\frac{8}{9\times 3-7\times 8}&\frac{9}{9\times 3-7\times 8}\end{matrix}\right)\left(\begin{matrix}6\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{29}&\frac{7}{29}\\\frac{8}{29}&-\frac{9}{29}\end{matrix}\right)\left(\begin{matrix}6\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{29}\times 6+\frac{7}{29}\times 9\\\frac{8}{29}\times 6-\frac{9}{29}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{45}{29}\\-\frac{33}{29}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{45}{29},y=-\frac{33}{29}$$

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.

$$9x+7y=6,8x+3y=9$$

Kia ōrite ai a 9x me 8x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 8 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 9.

$$8\times 9x+8\times 7y=8\times 6,9\times 8x+9\times 3y=9\times 9$$

Whakarūnātia.

$$72x+56y=48,72x+27y=81$$

Me tango 72x+27y=81 mai i 72x+56y=48 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$72x-72x+56y-27y=48-81$$

Tāpiri 72x ki te -72x. Ka whakakore atu ngā kupu 72x me -72x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$56y-27y=48-81$$

Tāpiri 56y ki te -27y.

$$29y=48-81$$

Tāpiri 48 ki te -81.

$$29y=-33$$

Whakawehea ngā taha e rua ki te 29.

$$y=-\frac{33}{29}$$

Whakaurua te -\frac{33}{29} mō y ki 8x+3y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$8x+3\left(-\frac{33}{29}\right)=9$$

Whakareatia 3 ki te -\frac{33}{29}.

$$8x-\frac{99}{29}=9$$

Me tāpiri \frac{99}{29} ki ngā taha e rua o te whārite.

$$8x=\frac{360}{29}$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{45}{29}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{45}{29},y=-\frac{33}{29}$$