Whakaoti i te {l}{y=-7x+3}{y=-x-3}

$$\left\{ \begin{array} { l } { y = - 7 x + 3 } \\ { y = - x - 3 } \end{array} \right.$$

Whakaoti mō y, x

$$x=1$$

$$y=-4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te 7x ki ngā taha e rua.

$$y+7x=3$$

Whakaarohia te whārite tuarua. Me tāpiri te x ki ngā taha e rua.

$$y+x=-3$$

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.

$$y+7x=3,y+x=-3$$

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

$$y+7x=3$$

Me tango 7x mai i ngā taha e rua o te whārite.

$$y=-7x+3$$

Whakakapia te -7x+3 mō te y ki tērā atu whārite, y+x=-3.

$$-7x+3+x=-3$$

Tāpiri -7x ki te x.

$$-6x+3=-3$$

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

$$-6x=-6$$

Whakawehea ngā taha e rua ki te -6.

$$x=1$$

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

$$y=-7+3$$

Tāpiri 3 ki te -7.

$$y=-4$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te 7x ki ngā taha e rua.

$$y+7x=3$$

Whakaarohia te whārite tuarua. Me tāpiri te x ki ngā taha e rua.

$$y+x=-3$$

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

$$y+7x=3,y+x=-3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&7\\1&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}3\\-3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&7\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&7\\1&1\end{matrix}\right))\left(\begin{matrix}1&7\\1&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&7\\1&1\end{matrix}\right))\left(\begin{matrix}3\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&7\\1&1\end{matrix}\right))\left(\begin{matrix}3\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&7\\1&1\end{matrix}\right))\left(\begin{matrix}3\\-3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-7}&-\frac{7}{1-7}\\-\frac{1}{1-7}&\frac{1}{1-7}\end{matrix}\right)\left(\begin{matrix}3\\-3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{6}&\frac{7}{6}\\\frac{1}{6}&-\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}3\\-3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{6}\times 3+\frac{7}{6}\left(-3\right)\\\frac{1}{6}\times 3-\frac{1}{6}\left(-3\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

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

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te 7x ki ngā taha e rua.

$$y+7x=3$$

Whakaarohia te whārite tuarua. Me tāpiri te x ki ngā taha e rua.

$$y+x=-3$$

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.

$$y+7x=3,y+x=-3$$

Me tango y+x=-3 mai i y+7x=3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y+7x-x=3+3$$

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.

$$7x-x=3+3$$

Tāpiri 7x ki te -x.

$$6x=3+3$$

Tāpiri 3 ki te 3.

$$6x=6$$

Whakawehea ngā taha e rua ki te 6.

$$x=1$$

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

$$y+1=-3$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$y=-4$$

Kua oti te pūnaha te whakatau.

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