Whakaoti i te {c}{2x+y=5}{-4x+6y=12}

$$\left. \begin{array} { c } { 2 x + y = 5 } \\ { - 4 x + 6 y = 12 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{9}{8} = 1\frac{1}{8} = 1.125$$

$$y = \frac{11}{4} = 2\frac{3}{4} = 2.75$$

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.

$$2x+y=5,-4x+6y=12$$

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

$$2x+y=5$$

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

$$2x=-y+5$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

Whakakapia te \frac{-y+5}{2} mō te x ki tērā atu whārite, -4x+6y=12.

$$-4\left(-\frac{1}{2}y+\frac{5}{2}\right)+6y=12$$

Whakareatia -4 ki te \frac{-y+5}{2}.

$$2y-10+6y=12$$

Tāpiri 2y ki te 6y.

$$8y-10=12$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$8y=22$$

Whakawehea ngā taha e rua ki te 8.

$$y=\frac{11}{4}$$

Whakaurua te \frac{11}{4} mō y ki x=-\frac{1}{2}y+\frac{5}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times \frac{11}{4}+\frac{5}{2}$$

Whakareatia -\frac{1}{2} ki te \frac{11}{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.

$$x=-\frac{11}{8}+\frac{5}{2}$$

Tāpiri \frac{5}{2} ki te -\frac{11}{8} 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.

$$x=\frac{9}{8}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{9}{8},y=\frac{11}{4}$$

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.

$$2x+y=5,-4x+6y=12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&1\\-4&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\12\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&1\\-4&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&1\\-4&6\end{matrix}\right))\left(\begin{matrix}2&1\\-4&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\-4&6\end{matrix}\right))\left(\begin{matrix}5\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\-4&6\end{matrix}\right))\left(\begin{matrix}5\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\-4&6\end{matrix}\right))\left(\begin{matrix}5\\12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{2\times 6-\left(-4\right)}&-\frac{1}{2\times 6-\left(-4\right)}\\-\frac{-4}{2\times 6-\left(-4\right)}&\frac{2}{2\times 6-\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}5\\12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8}&-\frac{1}{16}\\\frac{1}{4}&\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}5\\12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8}\times 5-\frac{1}{16}\times 12\\\frac{1}{4}\times 5+\frac{1}{8}\times 12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{8}\\\frac{11}{4}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{9}{8},y=\frac{11}{4}$$

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.

$$2x+y=5,-4x+6y=12$$

Kia ōrite ai a 2x 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 2.

$$-4\times 2x-4y=-4\times 5,2\left(-4\right)x+2\times 6y=2\times 12$$

Whakarūnātia.

$$-8x-4y=-20,-8x+12y=24$$

Me tango -8x+12y=24 mai i -8x-4y=-20 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-8x+8x-4y-12y=-20-24$$

Tāpiri -8x ki te 8x. Ka whakakore atu ngā kupu -8x me 8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-4y-12y=-20-24$$

Tāpiri -4y ki te -12y.

$$-16y=-20-24$$

Tāpiri -20 ki te -24.

$$-16y=-44$$

Whakawehea ngā taha e rua ki te -16.

$$y=\frac{11}{4}$$

Whakaurua te \frac{11}{4} mō y ki -4x+6y=12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-4x+6\times \frac{11}{4}=12$$

Whakareatia 6 ki te \frac{11}{4}.

$$-4x+\frac{33}{2}=12$$

Me tango \frac{33}{2} mai i ngā taha e rua o te whārite.

$$-4x=-\frac{9}{2}$$

Whakawehea ngā taha e rua ki te -4.

$$x=\frac{9}{8}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{9}{8},y=\frac{11}{4}$$