Whakaoti i te {l}{4x+2y=6}{3x-y=-8}

$$\left\{ \begin{array} { l } { 4 x + 2 y = 6 } \\ { 3 x - y = - 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-1$$

$$y=5$$

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.

$$4x+2y=6,3x-y=-8$$

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

$$4x+2y=6$$

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

$$4x=-2y+6$$

Whakawehea ngā taha e rua ki te 4.

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

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

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

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

$$3\left(-\frac{1}{2}y+\frac{3}{2}\right)-y=-8$$

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

$$-\frac{3}{2}y+\frac{9}{2}-y=-8$$

Tāpiri -\frac{3y}{2} ki te -y.

$$-\frac{5}{2}y+\frac{9}{2}=-8$$

Me tango \frac{9}{2} mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua o te whārite ki te -\frac{5}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=5$$

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

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

Whakareatia -\frac{1}{2} ki te 5.

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

Tāpiri \frac{3}{2} ki te -\frac{5}{2} 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.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=5$$

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.

$$4x+2y=6,3x-y=-8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&2\\3&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\-8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&2\\3&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&2\\3&-1\end{matrix}\right))\left(\begin{matrix}4&2\\3&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\3&-1\end{matrix}\right))\left(\begin{matrix}6\\-8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\3&-1\end{matrix}\right))\left(\begin{matrix}6\\-8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\3&-1\end{matrix}\right))\left(\begin{matrix}6\\-8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4\left(-1\right)-2\times 3}&-\frac{2}{4\left(-1\right)-2\times 3}\\-\frac{3}{4\left(-1\right)-2\times 3}&\frac{4}{4\left(-1\right)-2\times 3}\end{matrix}\right)\left(\begin{matrix}6\\-8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10}&\frac{1}{5}\\\frac{3}{10}&-\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}6\\-8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10}\times 6+\frac{1}{5}\left(-8\right)\\\frac{3}{10}\times 6-\frac{2}{5}\left(-8\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=5$$

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.

$$4x+2y=6,3x-y=-8$$

Kia ōrite ai a 4x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$3\times 4x+3\times 2y=3\times 6,4\times 3x+4\left(-1\right)y=4\left(-8\right)$$

Whakarūnātia.

$$12x+6y=18,12x-4y=-32$$

Me tango 12x-4y=-32 mai i 12x+6y=18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+6y+4y=18+32$$

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

$$6y+4y=18+32$$

Tāpiri 6y ki te 4y.

$$10y=18+32$$

Tāpiri 18 ki te 32.

$$10y=50$$

Whakawehea ngā taha e rua ki te 10.

$$y=5$$

Whakaurua te 5 mō y ki 3x-y=-8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-5=-8$$

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

$$3x=-3$$

Whakawehea ngā taha e rua ki te 3.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=5$$