Whakaoti i te {l}{(x-5)=3(y-5)}{(x+10)=2(y+10)}

$$\left. \begin{array} { l } { ( x - 5 ) = 3 ( y - 5 ) } \\ { ( x + 10 ) = 2 ( y + 10 ) } \end{array} \right.$$

Whakaoti mō x, y

$$x=50$$

$$y=20$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te y-5.

$$x-5=3y-15$$

Tangohia te 3y mai i ngā taha e rua.

$$x-5-3y=-15$$

Me tāpiri te 5 ki ngā taha e rua.

$$x-3y=-15+5$$

Tāpirihia te -15 ki te 5, ka -10.

$$x-3y=-10$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te y+10.

$$x+10=2y+20$$

Tangohia te 2y mai i ngā taha e rua.

$$x+10-2y=20$$

Tangohia te 10 mai i ngā taha e rua.

$$x-2y=20-10$$

Tangohia te 10 i te 20, ka 10.

$$x-2y=10$$

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.

$$x-3y=-10,x-2y=10$$

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

$$x-3y=-10$$

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

$$x=3y-10$$

Whakakapia te 3y-10 mō te x ki tērā atu whārite, x-2y=10.

$$3y-10-2y=10$$

Tāpiri 3y ki te -2y.

$$y-10=10$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$y=20$$

Whakaurua te 20 mō y ki x=3y-10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=3\times 20-10$$

Whakareatia 3 ki te 20.

$$x=60-10$$

Tāpiri -10 ki te 60.

$$x=50$$

Kua oti te pūnaha te whakatau.

$$x=50,y=20$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te y-5.

$$x-5=3y-15$$

Tangohia te 3y mai i ngā taha e rua.

$$x-5-3y=-15$$

Me tāpiri te 5 ki ngā taha e rua.

$$x-3y=-15+5$$

Tāpirihia te -15 ki te 5, ka -10.

$$x-3y=-10$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te y+10.

$$x+10=2y+20$$

Tangohia te 2y mai i ngā taha e rua.

$$x+10-2y=20$$

Tangohia te 10 mai i ngā taha e rua.

$$x-2y=20-10$$

Tangohia te 10 i te 20, ka 10.

$$x-2y=10$$

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

$$x-3y=-10,x-2y=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-3\\1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-10\\10\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-3\\1&-2\end{matrix}\right))\left(\begin{matrix}1&-3\\1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\1&-2\end{matrix}\right))\left(\begin{matrix}-10\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\1&-2\end{matrix}\right))\left(\begin{matrix}-10\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\1&-2\end{matrix}\right))\left(\begin{matrix}-10\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-\left(-3\right)}&-\frac{-3}{-2-\left(-3\right)}\\-\frac{1}{-2-\left(-3\right)}&\frac{1}{-2-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}-10\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2&3\\-1&1\end{matrix}\right)\left(\begin{matrix}-10\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2\left(-10\right)+3\times 10\\-\left(-10\right)+10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\20\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=50,y=20$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te y-5.

$$x-5=3y-15$$

Tangohia te 3y mai i ngā taha e rua.

$$x-5-3y=-15$$

Me tāpiri te 5 ki ngā taha e rua.

$$x-3y=-15+5$$

Tāpirihia te -15 ki te 5, ka -10.

$$x-3y=-10$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te y+10.

$$x+10=2y+20$$

Tangohia te 2y mai i ngā taha e rua.

$$x+10-2y=20$$

Tangohia te 10 mai i ngā taha e rua.

$$x-2y=20-10$$

Tangohia te 10 i te 20, ka 10.

$$x-2y=10$$

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.

$$x-3y=-10,x-2y=10$$

Me tango x-2y=10 mai i x-3y=-10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x-3y+2y=-10-10$$

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

$$-3y+2y=-10-10$$

Tāpiri -3y ki te 2y.

$$-y=-10-10$$

Tāpiri -10 ki te -10.

$$-y=-20$$

Whakawehea ngā taha e rua ki te -1.

$$y=20$$

Whakaurua te 20 mō y ki x-2y=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-2\times 20=10$$

Whakareatia -2 ki te 20.

$$x-40=10$$

Me tāpiri 40 ki ngā taha e rua o te whārite.

$$x=50$$

Kua oti te pūnaha te whakatau.

$$x=50,y=20$$