Whakaoti i te {l}{x-3=y}{37-3x=y}

$$\left. \begin{array} { l } { x - 3 = y } \\ { 37 - 3 x = y } \end{array} \right.$$

Whakaoti mō x, y

$$x=10$$

$$y=7$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$x-3-y=0$$

Me tāpiri te 3 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x-y=3$$

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

$$37-3x-y=0$$

Tangohia te 37 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-3x-y=-37$$

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.

$$x-y=3,-3x-y=-37$$

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

$$x-y=3$$

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

$$x=y+3$$

Whakakapia te y+3 mō te x ki tērā atu whārite, -3x-y=-37.

$$-3\left(y+3\right)-y=-37$$

Whakareatia -3 ki te y+3.

$$-3y-9-y=-37$$

Tāpiri -3y ki te -y.

$$-4y-9=-37$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$-4y=-28$$

Whakawehea ngā taha e rua ki te -4.

$$y=7$$

Whakaurua te 7 mō y ki x=y+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=7+3$$

Tāpiri 3 ki te 7.

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=7$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$x-3-y=0$$

Me tāpiri te 3 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x-y=3$$

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

$$37-3x-y=0$$

Tangohia te 37 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-3x-y=-37$$

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

$$x-y=3,-3x-y=-37$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\-3&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\-37\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-1\\-3&-1\end{matrix}\right))\left(\begin{matrix}1&-1\\-3&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\-3&-1\end{matrix}\right))\left(\begin{matrix}3\\-37\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\-3&-1\end{matrix}\right))\left(\begin{matrix}3\\-37\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\-3&-1\end{matrix}\right))\left(\begin{matrix}3\\-37\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-\left(-\left(-3\right)\right)}&-\frac{-1}{-1-\left(-\left(-3\right)\right)}\\-\frac{-3}{-1-\left(-\left(-3\right)\right)}&\frac{1}{-1-\left(-\left(-3\right)\right)}\end{matrix}\right)\left(\begin{matrix}3\\-37\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&-\frac{1}{4}\\-\frac{3}{4}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}3\\-37\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\times 3-\frac{1}{4}\left(-37\right)\\-\frac{3}{4}\times 3-\frac{1}{4}\left(-37\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\7\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=10,y=7$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$x-3-y=0$$

Me tāpiri te 3 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x-y=3$$

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

$$37-3x-y=0$$

Tangohia te 37 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-3x-y=-37$$

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.

$$x-y=3,-3x-y=-37$$

Me tango -3x-y=-37 mai i x-y=3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x+3x-y+y=3+37$$

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

$$x+3x=3+37$$

Tāpiri x ki te 3x.

$$4x=3+37$$

Tāpiri 3 ki te 37.

$$4x=40$$

Whakawehea ngā taha e rua ki te 4.

$$x=10$$

Whakaurua te 10 mō x ki -3x-y=-37. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-3\times 10-y=-37$$

Whakareatia -3 ki te 10.

$$-30-y=-37$$

Me tāpiri 30 ki ngā taha e rua o te whārite.

$$-y=-7$$

Whakawehea ngā taha e rua ki te -1.

$$y=7$$

Kua oti te pūnaha te whakatau.

$$x=10,y=7$$