Whakaoti i te {l}{x-3y=4}{5x+3y=-1}

$$\left\{ \begin{array} { l } { x - 3 y = 4 } \\ { 5 x + 3 y = - 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{1}{2}=0.5$$

$$y = -\frac{7}{6} = -1\frac{1}{6} \approx -1.166666667$$

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.

$$x-3y=4,5x+3y=-1$$

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

$$x-3y=4$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$x=3y+4$$

Whakakapia te 3y+4 mō te x ki tērā atu whārite, 5x+3y=-1.

$$5\left(3y+4\right)+3y=-1$$

Whakareatia 5 ki te 3y+4.

$$15y+20+3y=-1$$

Tāpiri 15y ki te 3y.

$$18y+20=-1$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$18y=-21$$

Whakawehea ngā taha e rua ki te 18.

$$y=-\frac{7}{6}$$

Whakaurua te -\frac{7}{6} mō y ki x=3y+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=3\left(-\frac{7}{6}\right)+4$$

Whakareatia 3 ki te -\frac{7}{6}.

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

Tāpiri 4 ki te -\frac{7}{2}.

$$x=\frac{1}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{2},y=-\frac{7}{6}$$

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.

$$x-3y=4,5x+3y=-1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-3\\5&3\end{matrix}\right).

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\5&3\end{matrix}\right))\left(\begin{matrix}4\\-1\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}&\frac{1}{6}\\-\frac{5}{18}&\frac{1}{18}\end{matrix}\right)\left(\begin{matrix}4\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\times 4+\frac{1}{6}\left(-1\right)\\-\frac{5}{18}\times 4+\frac{1}{18}\left(-1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1}{2},y=-\frac{7}{6}$$

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.

$$x-3y=4,5x+3y=-1$$

Kia ōrite ai a x 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 1.

$$5x+5\left(-3\right)y=5\times 4,5x+3y=-1$$

Whakarūnātia.

$$5x-15y=20,5x+3y=-1$$

Me tango 5x+3y=-1 mai i 5x-15y=20 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5x-5x-15y-3y=20+1$$

Tāpiri 5x ki te -5x. Ka whakakore atu ngā kupu 5x me -5x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-15y-3y=20+1$$

Tāpiri -15y ki te -3y.

$$-18y=20+1$$

Tāpiri 20 ki te 1.

$$-18y=21$$

Whakawehea ngā taha e rua ki te -18.

$$y=-\frac{7}{6}$$

Whakaurua te -\frac{7}{6} mō y ki 5x+3y=-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x+3\left(-\frac{7}{6}\right)=-1$$

Whakareatia 3 ki te -\frac{7}{6}.

$$5x-\frac{7}{2}=-1$$

Me tāpiri \frac{7}{2} ki ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{2},y=-\frac{7}{6}$$