Whakaoti i te {l}{6.5x+y=9}{1.6x+0.2y=13}

$$\left. \begin{array} { l } { 6.5 x + y = 9 } \\ { 1.6 x + 0.2 y = 13 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{112}{3} = 37\frac{1}{3} \approx 37.333333333$$

$$y = -\frac{701}{3} = -233\frac{2}{3} \approx -233.666666667$$

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.

$$6.5x+y=9,1.6x+0.2y=13$$

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

$$6.5x+y=9$$

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

$$6.5x=-y+9$$

Whakawehea ngā taha e rua o te whārite ki te 6.5, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{2}{13}\left(-y+9\right)$$

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

$$x=-\frac{2}{13}y+\frac{18}{13}$$

Whakakapia te \frac{-2y+18}{13} mō te x ki tērā atu whārite, 1.6x+0.2y=13.

$$1.6\left(-\frac{2}{13}y+\frac{18}{13}\right)+0.2y=13$$

Whakareatia 1.6 ki te \frac{-2y+18}{13}.

$$-\frac{16}{65}y+\frac{144}{65}+0.2y=13$$

Tāpiri -\frac{16y}{65} ki te \frac{y}{5}.

$$-\frac{3}{65}y+\frac{144}{65}=13$$

Me tango \frac{144}{65} mai i ngā taha e rua o te whārite.

$$-\frac{3}{65}y=\frac{701}{65}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{3}{65}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{701}{3}$$

Whakaurua te -\frac{701}{3} mō y ki x=-\frac{2}{13}y+\frac{18}{13}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{13}\left(-\frac{701}{3}\right)+\frac{18}{13}$$

Whakareatia -\frac{2}{13} ki te -\frac{701}{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.

$$x=\frac{1402}{39}+\frac{18}{13}$$

Tāpiri \frac{18}{13} ki te \frac{1402}{39} 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.

$$x=\frac{112}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{112}{3},y=-\frac{701}{3}$$

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.

$$6.5x+y=9,1.6x+0.2y=13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6.5&1\\1.6&0.2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\13\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6.5&1\\1.6&0.2\end{matrix}\right).

$$inverse(\left(\begin{matrix}6.5&1\\1.6&0.2\end{matrix}\right))\left(\begin{matrix}6.5&1\\1.6&0.2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6.5&1\\1.6&0.2\end{matrix}\right))\left(\begin{matrix}9\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6.5&1\\1.6&0.2\end{matrix}\right))\left(\begin{matrix}9\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6.5&1\\1.6&0.2\end{matrix}\right))\left(\begin{matrix}9\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{0.2}{6.5\times 0.2-1.6}&-\frac{1}{6.5\times 0.2-1.6}\\-\frac{1.6}{6.5\times 0.2-1.6}&\frac{6.5}{6.5\times 0.2-1.6}\end{matrix}\right)\left(\begin{matrix}9\\13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{3}&\frac{10}{3}\\\frac{16}{3}&-\frac{65}{3}\end{matrix}\right)\left(\begin{matrix}9\\13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{3}\times 9+\frac{10}{3}\times 13\\\frac{16}{3}\times 9-\frac{65}{3}\times 13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{112}{3}\\-\frac{701}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{112}{3},y=-\frac{701}{3}$$

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.

$$6.5x+y=9,1.6x+0.2y=13$$

Kia ōrite ai a \frac{13x}{2} me \frac{8x}{5}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1.6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 6.5.

$$1.6\times 6.5x+1.6y=1.6\times 9,6.5\times 1.6x+6.5\times 0.2y=6.5\times 13$$

Whakarūnātia.

$$10.4x+1.6y=14.4,10.4x+1.3y=84.5$$

Me tango 10.4x+1.3y=84.5 mai i 10.4x+1.6y=14.4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10.4x-10.4x+1.6y-1.3y=14.4-84.5$$

Tāpiri \frac{52x}{5} ki te -\frac{52x}{5}. Ka whakakore atu ngā kupu \frac{52x}{5} me -\frac{52x}{5}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$1.6y-1.3y=14.4-84.5$$

Tāpiri \frac{8y}{5} ki te -\frac{13y}{10}.

$$0.3y=14.4-84.5$$

Tāpiri 14.4 ki te -84.5 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.

$$0.3y=-70.1$$

Whakawehea ngā taha e rua o te whārite ki te 0.3, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{701}{3}$$

Whakaurua te -\frac{701}{3} mō y ki 1.6x+0.2y=13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$1.6x+0.2\left(-\frac{701}{3}\right)=13$$

Whakareatia 0.2 ki te -\frac{701}{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.

$$1.6x-\frac{701}{15}=13$$

Me tāpiri \frac{701}{15} ki ngā taha e rua o te whārite.

$$1.6x=\frac{896}{15}$$

Whakawehea ngā taha e rua o te whārite ki te 1.6, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{112}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{112}{3},y=-\frac{701}{3}$$