Whakaoti i te {l}{0,6x+2y=20}{(2)-4x+y=-1}

$$\left\{ \begin{array} { l } { 0,6 x + 2 y = 20 } \\ { ( 2 ) - 4 x + y = - 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{130}{43} = 3\frac{1}{43} \approx 3.023255814$$

$$y = \frac{391}{43} = 9\frac{4}{43} \approx 9.093023256$$

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.

$$0.6x+2y=20,-4x+y+2=-1$$

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

$$0.6x+2y=20$$

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

$$0.6x=-2y+20$$

Whakawehea ngā taha e rua o te whārite ki te 0.6, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{5}{3}\left(-2y+20\right)$$

Whakareatia \frac{5}{3} ki te -2y+20.

$$x=-\frac{10}{3}y+\frac{100}{3}$$

Whakakapia te \frac{-10y+100}{3} mō te x ki tērā atu whārite, -4x+y+2=-1.

$$-4\left(-\frac{10}{3}y+\frac{100}{3}\right)+y+2=-1$$

Whakareatia -4 ki te \frac{-10y+100}{3}.

$$\frac{40}{3}y-\frac{400}{3}+y+2=-1$$

Tāpiri \frac{40y}{3} ki te y.

$$\frac{43}{3}y-\frac{400}{3}+2=-1$$

Tāpiri -\frac{400}{3} ki te 2.

$$\frac{43}{3}y-\frac{394}{3}=-1$$

Me tāpiri \frac{394}{3} ki ngā taha e rua o te whārite.

$$\frac{43}{3}y=\frac{391}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{43}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{391}{43}$$

Whakaurua te \frac{391}{43} mō y ki x=-\frac{10}{3}y+\frac{100}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{10}{3}\times \frac{391}{43}+\frac{100}{3}$$

Whakareatia -\frac{10}{3} ki te \frac{391}{43} 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.

$$x=-\frac{3910}{129}+\frac{100}{3}$$

Tāpiri \frac{100}{3} ki te -\frac{3910}{129} 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.

$$x=\frac{130}{43}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{130}{43},y=\frac{391}{43}$$

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.

$$0.6x+2y=20,-4x+y+2=-1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}0.6&2\\-4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}20\\-3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}0.6&2\\-4&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}0.6&2\\-4&1\end{matrix}\right))\left(\begin{matrix}0.6&2\\-4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.6&2\\-4&1\end{matrix}\right))\left(\begin{matrix}20\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.6&2\\-4&1\end{matrix}\right))\left(\begin{matrix}20\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.6&2\\-4&1\end{matrix}\right))\left(\begin{matrix}20\\-3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{0.6-2\left(-4\right)}&-\frac{2}{0.6-2\left(-4\right)}\\-\frac{-4}{0.6-2\left(-4\right)}&\frac{0.6}{0.6-2\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}20\\-3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{43}&-\frac{10}{43}\\\frac{20}{43}&\frac{3}{43}\end{matrix}\right)\left(\begin{matrix}20\\-3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{43}\times 20-\frac{10}{43}\left(-3\right)\\\frac{20}{43}\times 20+\frac{3}{43}\left(-3\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{130}{43}\\\frac{391}{43}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{130}{43},y=\frac{391}{43}$$

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.

$$0.6x+2y=20,-4x+y+2=-1$$

Kia ōrite ai a \frac{3x}{5} me -4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 0.6.

$$-4\times 0.6x-4\times 2y=-4\times 20,0.6\left(-4\right)x+0.6y+0.6\times 2=0.6\left(-1\right)$$

Whakarūnātia.

$$-2.4x-8y=-80,-2.4x+0.6y+1.2=-0.6$$

Me tango -2.4x+0.6y+1.2=-0.6 mai i -2.4x-8y=-80 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2.4x+2.4x-8y-0.6y-1.2=-80+0.6$$

Tāpiri -\frac{12x}{5} ki te \frac{12x}{5}. Ka whakakore atu ngā kupu -\frac{12x}{5} me \frac{12x}{5}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-8y-0.6y-1.2=-80+0.6$$

Tāpiri -8y ki te -\frac{3y}{5}.

$$-8.6y-1.2=-80+0.6$$

Tāpiri -80 ki te 0.6.

$$-8.6y-1.2=-79.4$$

Me tāpiri 1.2 ki ngā taha e rua o te whārite.

$$-8.6y=-78.2$$

Whakawehea ngā taha e rua o te whārite ki te -8.6, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{391}{43}$$

Whakaurua te \frac{391}{43} mō y ki -4x+y+2=-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-4x+\frac{391}{43}+2=-1$$

Tāpiri \frac{391}{43} ki te 2.

$$-4x+\frac{477}{43}=-1$$

Me tango \frac{477}{43} mai i ngā taha e rua o te whārite.

$$-4x=-\frac{520}{43}$$

Whakawehea ngā taha e rua ki te -4.

$$x=\frac{130}{43}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{130}{43},y=\frac{391}{43}$$