Whakaoti i te {l}{x+2y=-8}{y=-3/2x-2}

$$\left. \begin{array} { l } { x + 2 y = - 8 } \\ { y = - \frac { 3 } { 2 } x - 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=-5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Me tāpiri te \frac{3}{2}x ki ngā taha e rua.

$$y+\frac{3}{2}x=-2$$

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.

$$x+2y=-8,\frac{3}{2}x+y=-2$$

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

$$x+2y=-8$$

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

$$x=-2y-8$$

Whakakapia te -2y-8 mō te x ki tērā atu whārite, \frac{3}{2}x+y=-2.

$$\frac{3}{2}\left(-2y-8\right)+y=-2$$

Whakareatia \frac{3}{2} ki te -2y-8.

$$-3y-12+y=-2$$

Tāpiri -3y ki te y.

$$-2y-12=-2$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$-2y=10$$

Whakawehea ngā taha e rua ki te -2.

$$y=-5$$

Whakaurua te -5 mō y ki x=-2y-8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\left(-5\right)-8$$

Whakareatia -2 ki te -5.

$$x=10-8$$

Tāpiri -8 ki te 10.

$$x=2$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Me tāpiri te \frac{3}{2}x ki ngā taha e rua.

$$y+\frac{3}{2}x=-2$$

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

$$x+2y=-8,\frac{3}{2}x+y=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\\frac{3}{2}&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-8\\-2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&2\\\frac{3}{2}&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&2\\\frac{3}{2}&1\end{matrix}\right))\left(\begin{matrix}1&2\\\frac{3}{2}&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\\frac{3}{2}&1\end{matrix}\right))\left(\begin{matrix}-8\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\\frac{3}{2}&1\end{matrix}\right))\left(\begin{matrix}-8\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\\frac{3}{2}&1\end{matrix}\right))\left(\begin{matrix}-8\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-2\times \frac{3}{2}}&-\frac{2}{1-2\times \frac{3}{2}}\\-\frac{\frac{3}{2}}{1-2\times \frac{3}{2}}&\frac{1}{1-2\times \frac{3}{2}}\end{matrix}\right)\left(\begin{matrix}-8\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}&1\\\frac{3}{4}&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}-8\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}\left(-8\right)-2\\\frac{3}{4}\left(-8\right)-\frac{1}{2}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Me tāpiri te \frac{3}{2}x ki ngā taha e rua.

$$y+\frac{3}{2}x=-2$$

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.

$$x+2y=-8,\frac{3}{2}x+y=-2$$

Kia ōrite ai a x me \frac{3x}{2}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te \frac{3}{2} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$\frac{3}{2}x+\frac{3}{2}\times 2y=\frac{3}{2}\left(-8\right),\frac{3}{2}x+y=-2$$

Whakarūnātia.

$$\frac{3}{2}x+3y=-12,\frac{3}{2}x+y=-2$$

Me tango \frac{3}{2}x+y=-2 mai i \frac{3}{2}x+3y=-12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{3}{2}x-\frac{3}{2}x+3y-y=-12+2$$

Tāpiri \frac{3x}{2} ki te -\frac{3x}{2}. Ka whakakore atu ngā kupu \frac{3x}{2} me -\frac{3x}{2}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3y-y=-12+2$$

Tāpiri 3y ki te -y.

$$2y=-12+2$$

Tāpiri -12 ki te 2.

$$2y=-10$$

Whakawehea ngā taha e rua ki te 2.

$$y=-5$$

Whakaurua te -5 mō y ki \frac{3}{2}x+y=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$\frac{3}{2}x-5=-2$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$\frac{3}{2}x=3$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=2$$

Kua oti te pūnaha te whakatau.

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