Whakaoti i te {r}{6x+y=4}{x-4y=19}

$$\left\{ \begin{array} { r } { 6 x + y = 4 } \\ { x - 4 y = 19 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{7}{5} = 1\frac{2}{5} = 1.4$$

$$y = -\frac{22}{5} = -4\frac{2}{5} = -4.4$$

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.

$$6x+y=4,x-4y=19$$

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

$$6x+y=4$$

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

$$6x=-y+4$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(-y+4\right)$$

Whakareatia \frac{1}{6} ki te -y+4.

$$x=-\frac{1}{6}y+\frac{2}{3}$$

Whakakapia te -\frac{y}{6}+\frac{2}{3} mō te x ki tērā atu whārite, x-4y=19.

$$-\frac{1}{6}y+\frac{2}{3}-4y=19$$

Tāpiri -\frac{y}{6} ki te -4y.

$$-\frac{25}{6}y+\frac{2}{3}=19$$

Me tango \frac{2}{3} mai i ngā taha e rua o te whārite.

$$-\frac{25}{6}y=\frac{55}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{25}{6}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{22}{5}$$

Whakaurua te -\frac{22}{5} mō y ki x=-\frac{1}{6}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.

$$x=-\frac{1}{6}\left(-\frac{22}{5}\right)+\frac{2}{3}$$

Whakareatia -\frac{1}{6} ki te -\frac{22}{5} 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.

$$x=\frac{11}{15}+\frac{2}{3}$$

Tāpiri \frac{2}{3} ki te \frac{11}{15} 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.

$$x=\frac{7}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{7}{5},y=-\frac{22}{5}$$

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.

$$6x+y=4,x-4y=19$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&1\\1&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\19\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&1\\1&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&1\\1&-4\end{matrix}\right))\left(\begin{matrix}6&1\\1&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&1\\1&-4\end{matrix}\right))\left(\begin{matrix}4\\19\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&1\\1&-4\end{matrix}\right))\left(\begin{matrix}4\\19\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&1\\1&-4\end{matrix}\right))\left(\begin{matrix}4\\19\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{6\left(-4\right)-1}&-\frac{1}{6\left(-4\right)-1}\\-\frac{1}{6\left(-4\right)-1}&\frac{6}{6\left(-4\right)-1}\end{matrix}\right)\left(\begin{matrix}4\\19\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{25}&\frac{1}{25}\\\frac{1}{25}&-\frac{6}{25}\end{matrix}\right)\left(\begin{matrix}4\\19\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{25}\times 4+\frac{1}{25}\times 19\\\frac{1}{25}\times 4-\frac{6}{25}\times 19\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{5}\\-\frac{22}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{7}{5},y=-\frac{22}{5}$$

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.

$$6x+y=4,x-4y=19$$

Kia ōrite ai a 6x 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 6.

$$6x+y=4,6x+6\left(-4\right)y=6\times 19$$

Whakarūnātia.

$$6x+y=4,6x-24y=114$$

Me tango 6x-24y=114 mai i 6x+y=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+y+24y=4-114$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$y+24y=4-114$$

Tāpiri y ki te 24y.

$$25y=4-114$$

Tāpiri 4 ki te -114.

$$25y=-110$$

Whakawehea ngā taha e rua ki te 25.

$$y=-\frac{22}{5}$$

Whakaurua te -\frac{22}{5} mō y ki x-4y=19. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-4\left(-\frac{22}{5}\right)=19$$

Whakareatia -4 ki te -\frac{22}{5}.

$$x+\frac{88}{5}=19$$

Me tango \frac{88}{5} mai i ngā taha e rua o te whārite.

$$x=\frac{7}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{7}{5},y=-\frac{22}{5}$$