Whakaoti i te {l}{4x-7y=23}{6x+2y=-3}

$$\left. \begin{array} { l } { 4 x - 7 y = 23 } \\ { 6 x + 2 y = - 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{1}{2}=0.5$$

$$y=-3$$

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.

$$4x-7y=23,6x+2y=-3$$

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

$$4x-7y=23$$

Me tāpiri 7y ki ngā taha e rua o te whārite.

$$4x=7y+23$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(7y+23\right)$$

Whakareatia \frac{1}{4} ki te 7y+23.

$$x=\frac{7}{4}y+\frac{23}{4}$$

Whakakapia te \frac{7y+23}{4} mō te x ki tērā atu whārite, 6x+2y=-3.

$$6\left(\frac{7}{4}y+\frac{23}{4}\right)+2y=-3$$

Whakareatia 6 ki te \frac{7y+23}{4}.

$$\frac{21}{2}y+\frac{69}{2}+2y=-3$$

Tāpiri \frac{21y}{2} ki te 2y.

$$\frac{25}{2}y+\frac{69}{2}=-3$$

Me tango \frac{69}{2} mai i ngā taha e rua o te whārite.

$$\frac{25}{2}y=-\frac{75}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{25}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-3$$

Whakaurua te -3 mō y ki x=\frac{7}{4}y+\frac{23}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{7}{4}\left(-3\right)+\frac{23}{4}$$

Whakareatia \frac{7}{4} ki te -3.

$$x=\frac{-21+23}{4}$$

Tāpiri \frac{23}{4} ki te -\frac{21}{4} 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.

$$x=\frac{1}{2}$$

Kua oti te pūnaha te whakatau.

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

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.

$$4x-7y=23,6x+2y=-3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-7\\6&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}23\\-3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-7\\6&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&-7\\6&2\end{matrix}\right))\left(\begin{matrix}4&-7\\6&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-7\\6&2\end{matrix}\right))\left(\begin{matrix}23\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-7\\6&2\end{matrix}\right))\left(\begin{matrix}23\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-7\\6&2\end{matrix}\right))\left(\begin{matrix}23\\-3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{4\times 2-\left(-7\times 6\right)}&-\frac{-7}{4\times 2-\left(-7\times 6\right)}\\-\frac{6}{4\times 2-\left(-7\times 6\right)}&\frac{4}{4\times 2-\left(-7\times 6\right)}\end{matrix}\right)\left(\begin{matrix}23\\-3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{25}&\frac{7}{50}\\-\frac{3}{25}&\frac{2}{25}\end{matrix}\right)\left(\begin{matrix}23\\-3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{25}\times 23+\frac{7}{50}\left(-3\right)\\-\frac{3}{25}\times 23+\frac{2}{25}\left(-3\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\\-3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

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

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.

$$4x-7y=23,6x+2y=-3$$

Kia ōrite ai a 4x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$6\times 4x+6\left(-7\right)y=6\times 23,4\times 6x+4\times 2y=4\left(-3\right)$$

Whakarūnātia.

$$24x-42y=138,24x+8y=-12$$

Me tango 24x+8y=-12 mai i 24x-42y=138 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$24x-24x-42y-8y=138+12$$

Tāpiri 24x ki te -24x. Ka whakakore atu ngā kupu 24x me -24x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-42y-8y=138+12$$

Tāpiri -42y ki te -8y.

$$-50y=138+12$$

Tāpiri 138 ki te 12.

$$-50y=150$$

Whakawehea ngā taha e rua ki te -50.

$$y=-3$$

Whakaurua te -3 mō y ki 6x+2y=-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x+2\left(-3\right)=-3$$

Whakareatia 2 ki te -3.

$$6x-6=-3$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$6x=3$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{2}$$

Kua oti te pūnaha te whakatau.

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