Whakaoti i te {l}{x+5=3y}{y-2=2x}

$$\left. \begin{array} { l } { x + 5 = 3 y } \\ { y - 2 = 2 x } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{1}{5}=-0.2$$

$$y = \frac{8}{5} = 1\frac{3}{5} = 1.6$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 3y mai i ngā taha e rua.

$$x+5-3y=0$$

Tangohia te 5 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x-3y=-5$$

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

$$y-2-2x=0$$

Me tāpiri te 2 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$y-2x=2$$

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.

$$x-3y=-5,-2x+y=2$$

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

$$x-3y=-5$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$x=3y-5$$

Whakakapia te 3y-5 mō te x ki tērā atu whārite, -2x+y=2.

$$-2\left(3y-5\right)+y=2$$

Whakareatia -2 ki te 3y-5.

$$-6y+10+y=2$$

Tāpiri -6y ki te y.

$$-5y+10=2$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$-5y=-8$$

Whakawehea ngā taha e rua ki te -5.

$$y=\frac{8}{5}$$

Whakaurua te \frac{8}{5} mō y ki x=3y-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=3\times \frac{8}{5}-5$$

Whakareatia 3 ki te \frac{8}{5}.

$$x=\frac{24}{5}-5$$

Tāpiri -5 ki te \frac{24}{5}.

$$x=-\frac{1}{5}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{1}{5},y=\frac{8}{5}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 3y mai i ngā taha e rua.

$$x+5-3y=0$$

Tangohia te 5 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x-3y=-5$$

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

$$y-2-2x=0$$

Me tāpiri te 2 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$y-2x=2$$

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

$$x-3y=-5,-2x+y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-3\\-2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\\2\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-3\\-2&1\end{matrix}\right))\left(\begin{matrix}1&-3\\-2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\-2&1\end{matrix}\right))\left(\begin{matrix}-5\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\-2&1\end{matrix}\right))\left(\begin{matrix}-5\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\-2&1\end{matrix}\right))\left(\begin{matrix}-5\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-\left(-3\left(-2\right)\right)}&-\frac{-3}{1-\left(-3\left(-2\right)\right)}\\-\frac{-2}{1-\left(-3\left(-2\right)\right)}&\frac{1}{1-\left(-3\left(-2\right)\right)}\end{matrix}\right)\left(\begin{matrix}-5\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}&-\frac{3}{5}\\-\frac{2}{5}&-\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}-5\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}\left(-5\right)-\frac{3}{5}\times 2\\-\frac{2}{5}\left(-5\right)-\frac{1}{5}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}\\\frac{8}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{1}{5},y=\frac{8}{5}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 3y mai i ngā taha e rua.

$$x+5-3y=0$$

Tangohia te 5 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x-3y=-5$$

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

$$y-2-2x=0$$

Me tāpiri te 2 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$y-2x=2$$

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.

$$x-3y=-5,-2x+y=2$$

Kia ōrite ai a x 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 1.

$$-2x-2\left(-3\right)y=-2\left(-5\right),-2x+y=2$$

Whakarūnātia.

$$-2x+6y=10,-2x+y=2$$

Me tango -2x+y=2 mai i -2x+6y=10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x+6y-y=10-2$$

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

$$6y-y=10-2$$

Tāpiri 6y ki te -y.

$$5y=10-2$$

Tāpiri 10 ki te -2.

$$5y=8$$

Whakawehea ngā taha e rua ki te 5.

$$y=\frac{8}{5}$$

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

$$-2x+\frac{8}{5}=2$$

Me tango \frac{8}{5} mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te -2.

$$x=-\frac{1}{5}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{1}{5},y=\frac{8}{5}$$