Whakaoti i te {c}{2x+4y=1}{5x-y=2}

$$\left. \begin{array} { c } { 2 x + 4 y = 1 } \\ { 5 x - y = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{9}{22}\approx 0.409090909$$

$$y=\frac{1}{22}\approx 0.045454545$$

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.

$$2x+4y=1,5x-y=2$$

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

$$2x+4y=1$$

Me tango 4y mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te 2.

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

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

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

Whakakapia te -2y+\frac{1}{2} mō te x ki tērā atu whārite, 5x-y=2.

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

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

$$-10y+\frac{5}{2}-y=2$$

Tāpiri -10y ki te -y.

$$-11y+\frac{5}{2}=2$$

Me tango \frac{5}{2} mai i ngā taha e rua o te whārite.

$$-11y=-\frac{1}{2}$$

Whakawehea ngā taha e rua ki te -11.

$$y=\frac{1}{22}$$

Whakaurua te \frac{1}{22} mō y ki x=-2y+\frac{1}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times \frac{1}{22}+\frac{1}{2}$$

Whakareatia -2 ki te \frac{1}{22}.

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

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

$$x=\frac{9}{22}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{9}{22},y=\frac{1}{22}$$

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.

$$2x+4y=1,5x-y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&4\\5&-1\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{22}&\frac{2}{11}\\\frac{5}{22}&-\frac{1}{11}\end{matrix}\right)\left(\begin{matrix}1\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{22}+\frac{2}{11}\times 2\\\frac{5}{22}-\frac{1}{11}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{22}\\\frac{1}{22}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{9}{22},y=\frac{1}{22}$$

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.

$$2x+4y=1,5x-y=2$$

Kia ōrite ai a 2x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$5\times 2x+5\times 4y=5,2\times 5x+2\left(-1\right)y=2\times 2$$

Whakarūnātia.

$$10x+20y=5,10x-2y=4$$

Me tango 10x-2y=4 mai i 10x+20y=5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x+20y+2y=5-4$$

Tāpiri 10x ki te -10x. Ka whakakore atu ngā kupu 10x me -10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$20y+2y=5-4$$

Tāpiri 20y ki te 2y.

$$22y=5-4$$

Tāpiri 5 ki te -4.

$$22y=1$$

Whakawehea ngā taha e rua ki te 22.

$$y=\frac{1}{22}$$

Whakaurua te \frac{1}{22} mō y ki 5x-y=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-\frac{1}{22}=2$$

Me tāpiri \frac{1}{22} ki ngā taha e rua o te whārite.

$$5x=\frac{45}{22}$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{9}{22}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{9}{22},y=\frac{1}{22}$$