Whakaoti i te {l}{6x-2y=4}{y=5x+3}

$$\left\{ \begin{array} { l } { 6 x - 2 y = 4 } \\ { y = 5 x + 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

$$y = -\frac{19}{2} = -9\frac{1}{2} = -9.5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 5x mai i ngā taha e rua.

$$y-5x=3$$

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.

$$6x-2y=4,-5x+y=3$$

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

$$6x-2y=4$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$6x=2y+4$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(2y+4\right)$$

Whakareatia \frac{1}{6} ki te 4+2y.

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

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

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

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

$$-\frac{5}{3}y-\frac{10}{3}+y=3$$

Tāpiri -\frac{5y}{3} ki te y.

$$-\frac{2}{3}y-\frac{10}{3}=3$$

Me tāpiri \frac{10}{3} ki ngā taha e rua o te whārite.

$$-\frac{2}{3}y=\frac{19}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{2}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{19}{2}$$

Whakaurua te -\frac{19}{2} mō y ki x=\frac{1}{3}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.

$$x=\frac{1}{3}\left(-\frac{19}{2}\right)+\frac{2}{3}$$

Whakareatia \frac{1}{3} ki te -\frac{19}{2} 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.

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

Tāpiri \frac{2}{3} ki te -\frac{19}{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.

$$x=-\frac{5}{2}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{5}{2},y=-\frac{19}{2}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 5x mai i ngā taha e rua.

$$y-5x=3$$

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

$$6x-2y=4,-5x+y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&-2\\-5&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&-2\\-5&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&-2\\-5&1\end{matrix}\right))\left(\begin{matrix}6&-2\\-5&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-2\\-5&1\end{matrix}\right))\left(\begin{matrix}4\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-2\\-5&1\end{matrix}\right))\left(\begin{matrix}4\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-2\\-5&1\end{matrix}\right))\left(\begin{matrix}4\\3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6-\left(-2\left(-5\right)\right)}&-\frac{-2}{6-\left(-2\left(-5\right)\right)}\\-\frac{-5}{6-\left(-2\left(-5\right)\right)}&\frac{6}{6-\left(-2\left(-5\right)\right)}\end{matrix}\right)\left(\begin{matrix}4\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&-\frac{1}{2}\\-\frac{5}{4}&-\frac{3}{2}\end{matrix}\right)\left(\begin{matrix}4\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\times 4-\frac{1}{2}\times 3\\-\frac{5}{4}\times 4-\frac{3}{2}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{5}{2},y=-\frac{19}{2}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 5x mai i ngā taha e rua.

$$y-5x=3$$

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.

$$6x-2y=4,-5x+y=3$$

Kia ōrite ai a 6x me -5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 6.

$$-5\times 6x-5\left(-2\right)y=-5\times 4,6\left(-5\right)x+6y=6\times 3$$

Whakarūnātia.

$$-30x+10y=-20,-30x+6y=18$$

Me tango -30x+6y=18 mai i -30x+10y=-20 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-30x+30x+10y-6y=-20-18$$

Tāpiri -30x ki te 30x. Ka whakakore atu ngā kupu -30x me 30x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$10y-6y=-20-18$$

Tāpiri 10y ki te -6y.

$$4y=-20-18$$

Tāpiri -20 ki te -18.

$$4y=-38$$

Whakawehea ngā taha e rua ki te 4.

$$y=-\frac{19}{2}$$

Whakaurua te -\frac{19}{2} mō y ki -5x+y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-5x-\frac{19}{2}=3$$

Me tāpiri \frac{19}{2} ki ngā taha e rua o te whārite.

$$-5x=\frac{25}{2}$$

Whakawehea ngā taha e rua ki te -5.

$$x=-\frac{5}{2}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{5}{2},y=-\frac{19}{2}$$