Whakaoti i te {l}{4x-y=14}{6x+y=16}

$$\left\{ \begin{array} { l } { 4 x - y = 14 } \\ { 6 x + y = 16 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=-2$$

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.

$$4x-y=14,6x+y=16$$

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

$$4x-y=14$$

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

$$4x=y+14$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(y+14\right)$$

Whakareatia \frac{1}{4} ki te y+14.

$$x=\frac{1}{4}y+\frac{7}{2}$$

Whakakapia te \frac{y}{4}+\frac{7}{2} mō te x ki tērā atu whārite, 6x+y=16.

$$6\left(\frac{1}{4}y+\frac{7}{2}\right)+y=16$$

Whakareatia 6 ki te \frac{y}{4}+\frac{7}{2}.

$$\frac{3}{2}y+21+y=16$$

Tāpiri \frac{3y}{2} ki te y.

$$\frac{5}{2}y+21=16$$

Me tango 21 mai i ngā taha e rua o te whārite.

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

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.

$$y=-2$$

Whakaurua te -2 mō y ki x=\frac{1}{4}y+\frac{7}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{4}\left(-2\right)+\frac{7}{2}$$

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

$$x=\frac{-1+7}{2}$$

Tāpiri \frac{7}{2} ki te -\frac{1}{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.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=-2$$

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.

$$4x-y=14,6x+y=16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-1\\6&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14\\16\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}4&-1\\6&1\end{matrix}\right))\left(\begin{matrix}4&-1\\6&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\6&1\end{matrix}\right))\left(\begin{matrix}14\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\6&1\end{matrix}\right))\left(\begin{matrix}14\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\6&1\end{matrix}\right))\left(\begin{matrix}14\\16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4-\left(-6\right)}&-\frac{-1}{4-\left(-6\right)}\\-\frac{6}{4-\left(-6\right)}&\frac{4}{4-\left(-6\right)}\end{matrix}\right)\left(\begin{matrix}14\\16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10}&\frac{1}{10}\\-\frac{3}{5}&\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}14\\16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10}\times 14+\frac{1}{10}\times 16\\-\frac{3}{5}\times 14+\frac{2}{5}\times 16\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=-2$$

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.

$$4x-y=14,6x+y=16$$

Kia ōrite ai a 4x 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 4.

$$6\times 4x+6\left(-1\right)y=6\times 14,4\times 6x+4y=4\times 16$$

Whakarūnātia.

$$24x-6y=84,24x+4y=64$$

Me tango 24x+4y=64 mai i 24x-6y=84 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$24x-24x-6y-4y=84-64$$

Tāpiri 24x ki te -24x. Ka whakakore atu ngā kupu 24x me -24x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-6y-4y=84-64$$

Tāpiri -6y ki te -4y.

$$-10y=84-64$$

Tāpiri 84 ki te -64.

$$-10y=20$$

Whakawehea ngā taha e rua ki te -10.

$$y=-2$$

Whakaurua te -2 mō y ki 6x+y=16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x-2=16$$

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

$$6x=18$$

Whakawehea ngā taha e rua ki te 6.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=-2$$