Whakaoti i te {c}{x=3y+2}{x-5=4(y-5)}

$$\left. \begin{array} { c } { x = 3 y + 2 } \\ { x - 5 = 4 ( y - 5 ) } \end{array} \right.$$

Whakaoti mō x, y

$$x=53$$

$$y=17$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$x-3y=2$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te y-5.

$$x-5=4y-20$$

Tangohia te 4y mai i ngā taha e rua.

$$x-5-4y=-20$$

Me tāpiri te 5 ki ngā taha e rua.

$$x-4y=-20+5$$

Tāpirihia te -20 ki te 5, ka -15.

$$x-4y=-15$$

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.

$$x-3y=2,x-4y=-15$$

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

$$x-3y=2$$

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

$$x=3y+2$$

Whakakapia te 3y+2 mō te x ki tērā atu whārite, x-4y=-15.

$$3y+2-4y=-15$$

Tāpiri 3y ki te -4y.

$$-y+2=-15$$

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

$$-y=-17$$

Whakawehea ngā taha e rua ki te -1.

$$y=17$$

Whakaurua te 17 mō y ki x=3y+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=3\times 17+2$$

Whakareatia 3 ki te 17.

$$x=51+2$$

Tāpiri 2 ki te 51.

$$x=53$$

Kua oti te pūnaha te whakatau.

$$x=53,y=17$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$x-3y=2$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te y-5.

$$x-5=4y-20$$

Tangohia te 4y mai i ngā taha e rua.

$$x-5-4y=-20$$

Me tāpiri te 5 ki ngā taha e rua.

$$x-4y=-20+5$$

Tāpirihia te -20 ki te 5, ka -15.

$$x-4y=-15$$

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

$$x-3y=2,x-4y=-15$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-3\\1&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\-15\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-3\\1&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-3\\1&-4\end{matrix}\right))\left(\begin{matrix}1&-3\\1&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\1&-4\end{matrix}\right))\left(\begin{matrix}2\\-15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\1&-4\end{matrix}\right))\left(\begin{matrix}2\\-15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\1&-4\end{matrix}\right))\left(\begin{matrix}2\\-15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{-4-\left(-3\right)}&-\frac{-3}{-4-\left(-3\right)}\\-\frac{1}{-4-\left(-3\right)}&\frac{1}{-4-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}2\\-15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4&-3\\1&-1\end{matrix}\right)\left(\begin{matrix}2\\-15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\times 2-3\left(-15\right)\\2-\left(-15\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}53\\17\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=53,y=17$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$x-3y=2$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te y-5.

$$x-5=4y-20$$

Tangohia te 4y mai i ngā taha e rua.

$$x-5-4y=-20$$

Me tāpiri te 5 ki ngā taha e rua.

$$x-4y=-20+5$$

Tāpirihia te -20 ki te 5, ka -15.

$$x-4y=-15$$

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.

$$x-3y=2,x-4y=-15$$

Me tango x-4y=-15 mai i x-3y=2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x-3y+4y=2+15$$

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.

$$-3y+4y=2+15$$

Tāpiri -3y ki te 4y.

$$y=2+15$$

Tāpiri 2 ki te 15.

$$y=17$$

Whakaurua te 17 mō y ki x-4y=-15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-4\times 17=-15$$

Whakareatia -4 ki te 17.

$$x-68=-15$$

Me tāpiri 68 ki ngā taha e rua o te whārite.

$$x=53$$

Kua oti te pūnaha te whakatau.

$$x=53,y=17$$