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

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

Whakaoti mō x, y

$$x=1$$

$$y=-4$$

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.

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

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

$$5x+y=1$$

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

$$5x=-y+1$$

Whakawehea ngā taha e rua ki te 5.

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

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

$$x=-\frac{1}{5}y+\frac{1}{5}$$

Whakakapia te \frac{-y+1}{5} mō te x ki tērā atu whārite, 3x+y=-1.

$$3\left(-\frac{1}{5}y+\frac{1}{5}\right)+y=-1$$

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

$$-\frac{3}{5}y+\frac{3}{5}+y=-1$$

Tāpiri -\frac{3y}{5} ki te y.

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

Me tango \frac{3}{5} mai i ngā taha e rua o te whārite.

$$\frac{2}{5}y=-\frac{8}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{2}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-4$$

Whakaurua te -4 mō y ki x=-\frac{1}{5}y+\frac{1}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{5}\left(-4\right)+\frac{1}{5}$$

Whakareatia -\frac{1}{5} ki te -4.

$$x=\frac{4+1}{5}$$

Tāpiri \frac{1}{5} ki te \frac{4}{5} 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.

$$x=1$$

Kua oti te pūnaha te whakatau.

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

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

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

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

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

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

$$5x-3x=1+1$$

Tāpiri 5x ki te -3x.

$$2x=1+1$$

Tāpiri 1 ki te 1.

$$2x=2$$

Whakawehea ngā taha e rua ki te 2.

$$x=1$$

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

$$3+y=-1$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$y=-4$$

Kua oti te pūnaha te whakatau.

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