Whakaoti i te {l}{5x-4y=19}{x+2y=7}

$$\left. \begin{array} { l } { 5 x - 4 y = 19 } \\ { x + 2 y = 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{33}{7} = 4\frac{5}{7} \approx 4.714285714$$

$$y = \frac{8}{7} = 1\frac{1}{7} \approx 1.142857143$$

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.

$$5x-4y=19,x+2y=7$$

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

$$5x-4y=19$$

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

$$5x=4y+19$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(4y+19\right)$$

Whakareatia \frac{1}{5} ki te 4y+19.

$$x=\frac{4}{5}y+\frac{19}{5}$$

Whakakapia te \frac{4y+19}{5} mō te x ki tērā atu whārite, x+2y=7.

$$\frac{4}{5}y+\frac{19}{5}+2y=7$$

Tāpiri \frac{4y}{5} ki te 2y.

$$\frac{14}{5}y+\frac{19}{5}=7$$

Me tango \frac{19}{5} mai i ngā taha e rua o te whārite.

$$\frac{14}{5}y=\frac{16}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{14}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{8}{7}$$

Whakaurua te \frac{8}{7} mō y ki x=\frac{4}{5}y+\frac{19}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{4}{5}\times \frac{8}{7}+\frac{19}{5}$$

Whakareatia \frac{4}{5} ki te \frac{8}{7} 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.

$$x=\frac{32}{35}+\frac{19}{5}$$

Tāpiri \frac{19}{5} ki te \frac{32}{35} 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.

$$x=\frac{33}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{33}{7},y=\frac{8}{7}$$

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.

$$5x-4y=19,x+2y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-4\\1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}19\\7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-4\\1&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-4\\1&2\end{matrix}\right))\left(\begin{matrix}5&-4\\1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\1&2\end{matrix}\right))\left(\begin{matrix}19\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\1&2\end{matrix}\right))\left(\begin{matrix}19\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\1&2\end{matrix}\right))\left(\begin{matrix}19\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5\times 2-\left(-4\right)}&-\frac{-4}{5\times 2-\left(-4\right)}\\-\frac{1}{5\times 2-\left(-4\right)}&\frac{5}{5\times 2-\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}19\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}&\frac{2}{7}\\-\frac{1}{14}&\frac{5}{14}\end{matrix}\right)\left(\begin{matrix}19\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}\times 19+\frac{2}{7}\times 7\\-\frac{1}{14}\times 19+\frac{5}{14}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{33}{7}\\\frac{8}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{33}{7},y=\frac{8}{7}$$

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.

$$5x-4y=19,x+2y=7$$

Kia ōrite ai a 5x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$5x-4y=19,5x+5\times 2y=5\times 7$$

Whakarūnātia.

$$5x-4y=19,5x+10y=35$$

Me tango 5x+10y=35 mai i 5x-4y=19 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5x-5x-4y-10y=19-35$$

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

$$-4y-10y=19-35$$

Tāpiri -4y ki te -10y.

$$-14y=19-35$$

Tāpiri 19 ki te -35.

$$-14y=-16$$

Whakawehea ngā taha e rua ki te -14.

$$y=\frac{8}{7}$$

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

$$x+2\times \frac{8}{7}=7$$

Whakareatia 2 ki te \frac{8}{7}.

$$x+\frac{16}{7}=7$$

Me tango \frac{16}{7} mai i ngā taha e rua o te whārite.

$$x=\frac{33}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{33}{7},y=\frac{8}{7}$$