Whakaoti i te y=x-9;quady=5-x

$$y = x - 9 ; \quad y = 5 - x$$

Whakaoti mō y, x

$$x=7$$

$$y=-2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-x=-9$$

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

$$y+x=5$$

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.

$$y-x=-9,y+x=5$$

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

$$y-x=-9$$

Me tāpiri x ki ngā taha e rua o te whārite.

$$y=x-9$$

Whakakapia te x-9 mō te y ki tērā atu whārite, y+x=5.

$$x-9+x=5$$

Tāpiri x ki te x.

$$2x-9=5$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$2x=14$$

Whakawehea ngā taha e rua ki te 2.

$$x=7$$

Whakaurua te 7 mō x ki y=x-9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=7-9$$

Tāpiri -9 ki te 7.

$$y=-2$$

Kua oti te pūnaha te whakatau.

$$y=-2,x=7$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-x=-9$$

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

$$y+x=5$$

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

$$y-x=-9,y+x=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\1&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-9\\5\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-1\\1&1\end{matrix}\right))\left(\begin{matrix}1&-1\\1&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&1\end{matrix}\right))\left(\begin{matrix}-9\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&1\end{matrix}\right))\left(\begin{matrix}-9\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&1\end{matrix}\right))\left(\begin{matrix}-9\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-\left(-1\right)}&-\frac{-1}{1-\left(-1\right)}\\-\frac{1}{1-\left(-1\right)}&\frac{1}{1-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}-9\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{2}\\-\frac{1}{2}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}-9\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\left(-9\right)+\frac{1}{2}\times 5\\-\frac{1}{2}\left(-9\right)+\frac{1}{2}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=-2,x=7$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-x=-9$$

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

$$y+x=5$$

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.

$$y-x=-9,y+x=5$$

Me tango y+x=5 mai i y-x=-9 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-x-x=-9-5$$

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.

$$-x-x=-9-5$$

Tāpiri -x ki te -x.

$$-2x=-9-5$$

Tāpiri -9 ki te -5.

$$-2x=-14$$

Whakawehea ngā taha e rua ki te -2.

$$x=7$$

Whakaurua te 7 mō x ki y+x=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y+7=5$$

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

$$y=-2$$

Kua oti te pūnaha te whakatau.

$$y=-2,x=7$$