Whakaoti i te {l}{6x-18y=-85}{24x-5y=-5}

$$\left\{ \begin{array} { l } { 6 x - 18 y = - 85 } \\ { 24 x - 5 y = - 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{5}{6}\approx 0.833333333$$

$$y=5$$

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.

$$6x-18y=-85,24x-5y=-5$$

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

$$6x-18y=-85$$

Me tāpiri 18y ki ngā taha e rua o te whārite.

$$6x=18y-85$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(18y-85\right)$$

Whakareatia \frac{1}{6} ki te 18y-85.

$$x=3y-\frac{85}{6}$$

Whakakapia te 3y-\frac{85}{6} mō te x ki tērā atu whārite, 24x-5y=-5.

$$24\left(3y-\frac{85}{6}\right)-5y=-5$$

Whakareatia 24 ki te 3y-\frac{85}{6}.

$$72y-340-5y=-5$$

Tāpiri 72y ki te -5y.

$$67y-340=-5$$

Me tāpiri 340 ki ngā taha e rua o te whārite.

$$67y=335$$

Whakawehea ngā taha e rua ki te 67.

$$y=5$$

Whakaurua te 5 mō y ki x=3y-\frac{85}{6}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=3\times 5-\frac{85}{6}$$

Whakareatia 3 ki te 5.

$$x=15-\frac{85}{6}$$

Tāpiri -\frac{85}{6} ki te 15.

$$x=\frac{5}{6}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{6},y=5$$

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.

$$6x-18y=-85,24x-5y=-5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&-18\\24&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-85\\-5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&-18\\24&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&-18\\24&-5\end{matrix}\right))\left(\begin{matrix}6&-18\\24&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-18\\24&-5\end{matrix}\right))\left(\begin{matrix}-85\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-18\\24&-5\end{matrix}\right))\left(\begin{matrix}-85\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-18\\24&-5\end{matrix}\right))\left(\begin{matrix}-85\\-5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{6\left(-5\right)-\left(-18\times 24\right)}&-\frac{-18}{6\left(-5\right)-\left(-18\times 24\right)}\\-\frac{24}{6\left(-5\right)-\left(-18\times 24\right)}&\frac{6}{6\left(-5\right)-\left(-18\times 24\right)}\end{matrix}\right)\left(\begin{matrix}-85\\-5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{402}&\frac{3}{67}\\-\frac{4}{67}&\frac{1}{67}\end{matrix}\right)\left(\begin{matrix}-85\\-5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{402}\left(-85\right)+\frac{3}{67}\left(-5\right)\\-\frac{4}{67}\left(-85\right)+\frac{1}{67}\left(-5\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{6}\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{5}{6},y=5$$

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.

$$6x-18y=-85,24x-5y=-5$$

Kia ōrite ai a 6x me 24x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 24 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 6.

$$24\times 6x+24\left(-18\right)y=24\left(-85\right),6\times 24x+6\left(-5\right)y=6\left(-5\right)$$

Whakarūnātia.

$$144x-432y=-2040,144x-30y=-30$$

Me tango 144x-30y=-30 mai i 144x-432y=-2040 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$144x-144x-432y+30y=-2040+30$$

Tāpiri 144x ki te -144x. Ka whakakore atu ngā kupu 144x me -144x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-432y+30y=-2040+30$$

Tāpiri -432y ki te 30y.

$$-402y=-2040+30$$

Tāpiri -2040 ki te 30.

$$-402y=-2010$$

Whakawehea ngā taha e rua ki te -402.

$$y=5$$

Whakaurua te 5 mō y ki 24x-5y=-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$24x-5\times 5=-5$$

Whakareatia -5 ki te 5.

$$24x-25=-5$$

Me tāpiri 25 ki ngā taha e rua o te whārite.

$$24x=20$$

Whakawehea ngā taha e rua ki te 24.

$$x=\frac{5}{6}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{6},y=5$$