Whakaoti i te {l}{x+y=16}{10x+y=10y+x-18}

$$\left. \begin{array} { l } { x + y = 16 } \\ { 10 x + y = 10 y + x - 18 } \end{array} \right.$$

Whakaoti mō x, y

$$x=7$$

$$y=9$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 10y mai i ngā taha e rua.

$$10x+y-10y=x-18$$

Pahekotia te y me -10y, ka -9y.

$$10x-9y=x-18$$

Tangohia te x mai i ngā taha e rua.

$$10x-9y-x=-18$$

Pahekotia te 10x me -x, ka 9x.

$$9x-9y=-18$$

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.

$$x+y=16,9x-9y=-18$$

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

$$x+y=16$$

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

$$x=-y+16$$

Whakakapia te -y+16 mō te x ki tērā atu whārite, 9x-9y=-18.

$$9\left(-y+16\right)-9y=-18$$

Whakareatia 9 ki te -y+16.

$$-9y+144-9y=-18$$

Tāpiri -9y ki te -9y.

$$-18y+144=-18$$

Me tango 144 mai i ngā taha e rua o te whārite.

$$-18y=-162$$

Whakawehea ngā taha e rua ki te -18.

$$y=9$$

Whakaurua te 9 mō y ki x=-y+16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-9+16$$

Tāpiri 16 ki te -9.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=9$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 10y mai i ngā taha e rua.

$$10x+y-10y=x-18$$

Pahekotia te y me -10y, ka -9y.

$$10x-9y=x-18$$

Tangohia te x mai i ngā taha e rua.

$$10x-9y-x=-18$$

Pahekotia te 10x me -x, ka 9x.

$$9x-9y=-18$$

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

$$x+y=16,9x-9y=-18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\9&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}16\\-18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\9&-9\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\9&-9\end{matrix}\right))\left(\begin{matrix}1&1\\9&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\9&-9\end{matrix}\right))\left(\begin{matrix}16\\-18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\9&-9\end{matrix}\right))\left(\begin{matrix}16\\-18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\9&-9\end{matrix}\right))\left(\begin{matrix}16\\-18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{-9-9}&-\frac{1}{-9-9}\\-\frac{9}{-9-9}&\frac{1}{-9-9}\end{matrix}\right)\left(\begin{matrix}16\\-18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{18}\\\frac{1}{2}&-\frac{1}{18}\end{matrix}\right)\left(\begin{matrix}16\\-18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 16+\frac{1}{18}\left(-18\right)\\\frac{1}{2}\times 16-\frac{1}{18}\left(-18\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\9\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=7,y=9$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 10y mai i ngā taha e rua.

$$10x+y-10y=x-18$$

Pahekotia te y me -10y, ka -9y.

$$10x-9y=x-18$$

Tangohia te x mai i ngā taha e rua.

$$10x-9y-x=-18$$

Pahekotia te 10x me -x, ka 9x.

$$9x-9y=-18$$

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.

$$x+y=16,9x-9y=-18$$

Kia ōrite ai a x me 9x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 9 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$9x+9y=9\times 16,9x-9y=-18$$

Whakarūnātia.

$$9x+9y=144,9x-9y=-18$$

Me tango 9x-9y=-18 mai i 9x+9y=144 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$9x-9x+9y+9y=144+18$$

Tāpiri 9x ki te -9x. Ka whakakore atu ngā kupu 9x me -9x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$9y+9y=144+18$$

Tāpiri 9y ki te 9y.

$$18y=144+18$$

Tāpiri 144 ki te 18.

$$18y=162$$

Whakawehea ngā taha e rua ki te 18.

$$y=9$$

Whakaurua te 9 mō y ki 9x-9y=-18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$9x-9\times 9=-18$$

Whakareatia -9 ki te 9.

$$9x-81=-18$$

Me tāpiri 81 ki ngā taha e rua o te whārite.

$$9x=63$$

Whakawehea ngā taha e rua ki te 9.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=9$$