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

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

Whakaoti mō x, y

$$x=-1$$

$$y=-3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 5x mai i ngā taha e rua.

$$y-5x=2$$

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.

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

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

$$x+y=-4$$

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

$$x=-y-4$$

Whakakapia te -y-4 mō te x ki tērā atu whārite, -5x+y=2.

$$-5\left(-y-4\right)+y=2$$

Whakareatia -5 ki te -y-4.

$$5y+20+y=2$$

Tāpiri 5y ki te y.

$$6y+20=2$$

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

$$6y=-18$$

Whakawehea ngā taha e rua ki te 6.

$$y=-3$$

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

$$x=-\left(-3\right)-4$$

Whakareatia -1 ki te -3.

$$x=3-4$$

Tāpiri -4 ki te 3.

$$x=-1$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 5x mai i ngā taha e rua.

$$y-5x=2$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}&-\frac{1}{6}\\\frac{5}{6}&\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}-4\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\left(-4\right)-\frac{1}{6}\times 2\\\frac{5}{6}\left(-4\right)+\frac{1}{6}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 5x mai i ngā taha e rua.

$$y-5x=2$$

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.

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

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

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

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.

$$x+5x=-4-2$$

Tāpiri x ki te 5x.

$$6x=-4-2$$

Tāpiri -4 ki te -2.

$$6x=-6$$

Whakawehea ngā taha e rua ki te 6.

$$x=-1$$

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

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

Whakareatia -5 ki te -1.

$$5+y=2$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$y=-3$$

Kua oti te pūnaha te whakatau.

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