Whakaoti i te {l}{2x+10=4y-16x}{10y-10x=11y-12x}

$$\left\{ \begin{array} { l } { 2 x + 10 = 4 y - 16 x } \\ { 10 y - 10 x = 11 y - 12 x } \end{array} \right.$$

Whakaoti mō x, y

$$x=-1$$

$$y=-2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 4y mai i ngā taha e rua.

$$2x+10-4y=-16x$$

Me tāpiri te 16x ki ngā taha e rua.

$$2x+10-4y+16x=0$$

Pahekotia te 2x me 16x, ka 18x.

$$18x+10-4y=0$$

Tangohia te 10 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$18x-4y=-10$$

Whakaarohia te whārite tuarua. Tangohia te 11y mai i ngā taha e rua.

$$10y-10x-11y=-12x$$

Pahekotia te 10y me -11y, ka -y.

$$-y-10x=-12x$$

Me tāpiri te 12x ki ngā taha e rua.

$$-y-10x+12x=0$$

Pahekotia te -10x me 12x, ka 2x.

$$-y+2x=0$$

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.

$$18x-4y=-10,2x-y=0$$

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

$$18x-4y=-10$$

Me tāpiri 4y ki ngā taha e rua o te whārite.

$$18x=4y-10$$

Whakawehea ngā taha e rua ki te 18.

$$x=\frac{1}{18}\left(4y-10\right)$$

Whakareatia \frac{1}{18} ki te 4y-10.

$$x=\frac{2}{9}y-\frac{5}{9}$$

Whakakapia te \frac{2y-5}{9} mō te x ki tērā atu whārite, 2x-y=0.

$$2\left(\frac{2}{9}y-\frac{5}{9}\right)-y=0$$

Whakareatia 2 ki te \frac{2y-5}{9}.

$$\frac{4}{9}y-\frac{10}{9}-y=0$$

Tāpiri \frac{4y}{9} ki te -y.

$$-\frac{5}{9}y-\frac{10}{9}=0$$

Me tāpiri \frac{10}{9} ki ngā taha e rua o te whārite.

$$-\frac{5}{9}y=\frac{10}{9}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{5}{9}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-2$$

Whakaurua te -2 mō y ki x=\frac{2}{9}y-\frac{5}{9}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{9}\left(-2\right)-\frac{5}{9}$$

Whakareatia \frac{2}{9} ki te -2.

$$x=\frac{-4-5}{9}$$

Tāpiri -\frac{5}{9} ki te -\frac{4}{9} 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.

$$x=-1,y=-2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 4y mai i ngā taha e rua.

$$2x+10-4y=-16x$$

Me tāpiri te 16x ki ngā taha e rua.

$$2x+10-4y+16x=0$$

Pahekotia te 2x me 16x, ka 18x.

$$18x+10-4y=0$$

Tangohia te 10 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$18x-4y=-10$$

Whakaarohia te whārite tuarua. Tangohia te 11y mai i ngā taha e rua.

$$10y-10x-11y=-12x$$

Pahekotia te 10y me -11y, ka -y.

$$-y-10x=-12x$$

Me tāpiri te 12x ki ngā taha e rua.

$$-y-10x+12x=0$$

Pahekotia te -10x me 12x, ka 2x.

$$-y+2x=0$$

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

$$18x-4y=-10,2x-y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}18&-4\\2&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-10\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}18&-4\\2&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}18&-4\\2&-1\end{matrix}\right))\left(\begin{matrix}18&-4\\2&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}18&-4\\2&-1\end{matrix}\right))\left(\begin{matrix}-10\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}18&-4\\2&-1\end{matrix}\right))\left(\begin{matrix}-10\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}18&-4\\2&-1\end{matrix}\right))\left(\begin{matrix}-10\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{18\left(-1\right)-\left(-4\times 2\right)}&-\frac{-4}{18\left(-1\right)-\left(-4\times 2\right)}\\-\frac{2}{18\left(-1\right)-\left(-4\times 2\right)}&\frac{18}{18\left(-1\right)-\left(-4\times 2\right)}\end{matrix}\right)\left(\begin{matrix}-10\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10}&-\frac{2}{5}\\\frac{1}{5}&-\frac{9}{5}\end{matrix}\right)\left(\begin{matrix}-10\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10}\left(-10\right)\\\frac{1}{5}\left(-10\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=-2$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 4y mai i ngā taha e rua.

$$2x+10-4y=-16x$$

Me tāpiri te 16x ki ngā taha e rua.

$$2x+10-4y+16x=0$$

Pahekotia te 2x me 16x, ka 18x.

$$18x+10-4y=0$$

Tangohia te 10 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$18x-4y=-10$$

Whakaarohia te whārite tuarua. Tangohia te 11y mai i ngā taha e rua.

$$10y-10x-11y=-12x$$

Pahekotia te 10y me -11y, ka -y.

$$-y-10x=-12x$$

Me tāpiri te 12x ki ngā taha e rua.

$$-y-10x+12x=0$$

Pahekotia te -10x me 12x, ka 2x.

$$-y+2x=0$$

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.

$$18x-4y=-10,2x-y=0$$

Kia ōrite ai a 18x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 18.

$$2\times 18x+2\left(-4\right)y=2\left(-10\right),18\times 2x+18\left(-1\right)y=0$$

Whakarūnātia.

$$36x-8y=-20,36x-18y=0$$

Me tango 36x-18y=0 mai i 36x-8y=-20 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$36x-36x-8y+18y=-20$$

Tāpiri 36x ki te -36x. Ka whakakore atu ngā kupu 36x me -36x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-8y+18y=-20$$

Tāpiri -8y ki te 18y.

$$10y=-20$$

Whakawehea ngā taha e rua ki te 10.

$$y=-2$$

Whakaurua te -2 mō y ki 2x-y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-\left(-2\right)=0$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$2x=-2$$

Whakawehea ngā taha e rua ki te 2.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=-2$$