Whakaoti i te {l}{2x+y=6}{4x-y=7}

$$\left\{ \begin{array} { l } { 2 x + y = 6 } \\ { 4 x - y = 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{13}{6} = 2\frac{1}{6} \approx 2.166666667$$

$$y = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

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.

$$2x+y=6,4x-y=7$$

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

$$2x+y=6$$

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

$$2x=-y+6$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

Whakakapia te -\frac{y}{2}+3 mō te x ki tērā atu whārite, 4x-y=7.

$$4\left(-\frac{1}{2}y+3\right)-y=7$$

Whakareatia 4 ki te -\frac{y}{2}+3.

$$-2y+12-y=7$$

Tāpiri -2y ki te -y.

$$-3y+12=7$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$-3y=-5$$

Whakawehea ngā taha e rua ki te -3.

$$y=\frac{5}{3}$$

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

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

Whakareatia -\frac{1}{2} ki te \frac{5}{3} 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.

$$x=-\frac{5}{6}+3$$

Tāpiri 3 ki te -\frac{5}{6}.

$$x=\frac{13}{6}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{13}{6},y=\frac{5}{3}$$

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.

$$2x+y=6,4x-y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&1\\4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\7\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&1\\4&-1\end{matrix}\right))\left(\begin{matrix}2&1\\4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\4&-1\end{matrix}\right))\left(\begin{matrix}6\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\4&-1\end{matrix}\right))\left(\begin{matrix}6\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\4&-1\end{matrix}\right))\left(\begin{matrix}6\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2\left(-1\right)-4}&-\frac{1}{2\left(-1\right)-4}\\-\frac{4}{2\left(-1\right)-4}&\frac{2}{2\left(-1\right)-4}\end{matrix}\right)\left(\begin{matrix}6\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}&\frac{1}{6}\\\frac{2}{3}&-\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}6\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\times 6+\frac{1}{6}\times 7\\\frac{2}{3}\times 6-\frac{1}{3}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{6}\\\frac{5}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{13}{6},y=\frac{5}{3}$$

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.

$$2x+y=6,4x-y=7$$

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.

$$4\times 2x+4y=4\times 6,2\times 4x+2\left(-1\right)y=2\times 7$$

Whakarūnātia.

$$8x+4y=24,8x-2y=14$$

Me tango 8x-2y=14 mai i 8x+4y=24 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+4y+2y=24-14$$

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.

$$4y+2y=24-14$$

Tāpiri 4y ki te 2y.

$$6y=24-14$$

Tāpiri 24 ki te -14.

$$6y=10$$

Whakawehea ngā taha e rua ki te 6.

$$y=\frac{5}{3}$$

Whakaurua te \frac{5}{3} mō y ki 4x-y=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-\frac{5}{3}=7$$

Me tāpiri \frac{5}{3} ki ngā taha e rua o te whārite.

$$4x=\frac{26}{3}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{13}{6}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{13}{6},y=\frac{5}{3}$$