Whakaoti i te {l}{6x=-4y+5}{8y=6x-2}

$$\left. \begin{array} { l } { 6 x = - 4 y + 5 } \\ { 8 y = 6 x - 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{2}{3}\approx 0.666666667$$

$$y=\frac{1}{4}=0.25$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$6x+4y=5$$

Whakaarohia te whārite tuarua. Tangohia te 6x mai i ngā taha e rua.

$$8y-6x=-2$$

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.

$$6x+4y=5,-6x+8y=-2$$

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

$$6x+4y=5$$

Me tango 4y mai i ngā taha e rua o te whārite.

$$6x=-4y+5$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(-4y+5\right)$$

Whakareatia \frac{1}{6} ki te -4y+5.

$$x=-\frac{2}{3}y+\frac{5}{6}$$

Whakakapia te -\frac{2y}{3}+\frac{5}{6} mō te x ki tērā atu whārite, -6x+8y=-2.

$$-6\left(-\frac{2}{3}y+\frac{5}{6}\right)+8y=-2$$

Whakareatia -6 ki te -\frac{2y}{3}+\frac{5}{6}.

$$4y-5+8y=-2$$

Tāpiri 4y ki te 8y.

$$12y-5=-2$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$12y=3$$

Whakawehea ngā taha e rua ki te 12.

$$y=\frac{1}{4}$$

Whakaurua te \frac{1}{4} mō y ki x=-\frac{2}{3}y+\frac{5}{6}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{3}\times \frac{1}{4}+\frac{5}{6}$$

Whakareatia -\frac{2}{3} ki te \frac{1}{4} 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.

$$x=\frac{-1+5}{6}$$

Tāpiri \frac{5}{6} ki te -\frac{1}{6} 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.

$$x=\frac{2}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{2}{3},y=\frac{1}{4}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$6x+4y=5$$

Whakaarohia te whārite tuarua. Tangohia te 6x mai i ngā taha e rua.

$$8y-6x=-2$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$6x+4y=5,-6x+8y=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&4\\-6&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\-2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&4\\-6&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&4\\-6&8\end{matrix}\right))\left(\begin{matrix}6&4\\-6&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&4\\-6&8\end{matrix}\right))\left(\begin{matrix}5\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&4\\-6&8\end{matrix}\right))\left(\begin{matrix}5\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&4\\-6&8\end{matrix}\right))\left(\begin{matrix}5\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{6\times 8-4\left(-6\right)}&-\frac{4}{6\times 8-4\left(-6\right)}\\-\frac{-6}{6\times 8-4\left(-6\right)}&\frac{6}{6\times 8-4\left(-6\right)}\end{matrix}\right)\left(\begin{matrix}5\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}&-\frac{1}{18}\\\frac{1}{12}&\frac{1}{12}\end{matrix}\right)\left(\begin{matrix}5\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}\times 5-\frac{1}{18}\left(-2\right)\\\frac{1}{12}\times 5+\frac{1}{12}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}\\\frac{1}{4}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{2}{3},y=\frac{1}{4}$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$6x+4y=5$$

Whakaarohia te whārite tuarua. Tangohia te 6x mai i ngā taha e rua.

$$8y-6x=-2$$

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.

$$6x+4y=5,-6x+8y=-2$$

Kia ōrite ai a 6x 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 6.

$$-6\times 6x-6\times 4y=-6\times 5,6\left(-6\right)x+6\times 8y=6\left(-2\right)$$

Whakarūnātia.

$$-36x-24y=-30,-36x+48y=-12$$

Me tango -36x+48y=-12 mai i -36x-24y=-30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-36x+36x-24y-48y=-30+12$$

Tāpiri -36x ki te 36x. Ka whakakore atu ngā kupu -36x me 36x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-24y-48y=-30+12$$

Tāpiri -24y ki te -48y.

$$-72y=-30+12$$

Tāpiri -30 ki te 12.

$$-72y=-18$$

Whakawehea ngā taha e rua ki te -72.

$$y=\frac{1}{4}$$

Whakaurua te \frac{1}{4} mō y ki -6x+8y=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-6x+8\times \frac{1}{4}=-2$$

Whakareatia 8 ki te \frac{1}{4}.

$$-6x+2=-2$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$-6x=-4$$

Whakawehea ngā taha e rua ki te -6.

$$x=\frac{2}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{2}{3},y=\frac{1}{4}$$