Whakaoti i te {l}{4x-y=1}{2x+y=4}

$$\left. \begin{array} { l } { 4 x - y = 1 } \\ { 2 x + y = 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{5}{6}\approx 0.833333333$$

$$y = \frac{7}{3} = 2\frac{1}{3} \approx 2.333333333$$

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.

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

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

$$4x-y=1$$

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

$$4x=y+1$$

Whakawehea ngā taha e rua ki te 4.

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

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

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

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

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

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

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

Tāpiri \frac{y}{2} ki te y.

$$\frac{3}{2}y+\frac{1}{2}=4$$

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$\frac{3}{2}y=\frac{7}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{7}{3}$$

Whakaurua te \frac{7}{3} mō y ki x=\frac{1}{4}y+\frac{1}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{4}\times \frac{7}{3}+\frac{1}{4}$$

Whakareatia \frac{1}{4} ki te \frac{7}{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.

$$x=\frac{7}{12}+\frac{1}{4}$$

Tāpiri \frac{1}{4} ki te \frac{7}{12} 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.

$$x=\frac{5}{6}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{6},y=\frac{7}{3}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{6}\\\frac{7}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{5}{6},y=\frac{7}{3}$$

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.

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

Kia ōrite ai a 4x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

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

Whakarūnātia.

$$8x-2y=2,8x+4y=16$$

Me tango 8x+4y=16 mai i 8x-2y=2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x-2y-4y=2-16$$

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.

$$-2y-4y=2-16$$

Tāpiri -2y ki te -4y.

$$-6y=2-16$$

Tāpiri 2 ki te -16.

$$-6y=-14$$

Whakawehea ngā taha e rua ki te -6.

$$y=\frac{7}{3}$$

Whakaurua te \frac{7}{3} mō y ki 2x+y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+\frac{7}{3}=4$$

Me tango \frac{7}{3} mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{5}{6}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{6},y=\frac{7}{3}$$