Whakaoti i te {l}{x=4+y}{2x+3=y}

$$\left. \begin{array} { l } { x = 4 + y } \\ { 2 x + 3 = y } \end{array} \right.$$

Whakaoti mō x, y

$$x=-7$$

$$y=-11$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$x-y=4$$

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

$$2x+3-y=0$$

Tangohia te 3 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$2x-y=-3$$

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.

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

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

$$x-y=4$$

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

$$x=y+4$$

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

$$2\left(y+4\right)-y=-3$$

Whakareatia 2 ki te y+4.

$$2y+8-y=-3$$

Tāpiri 2y ki te -y.

$$y+8=-3$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$y=-11$$

Whakaurua te -11 mō y ki x=y+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-11+4$$

Tāpiri 4 ki te -11.

$$x=-7$$

Kua oti te pūnaha te whakatau.

$$x=-7,y=-11$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$x-y=4$$

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

$$2x+3-y=0$$

Tangohia te 3 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$2x-y=-3$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-7\\-11\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-7,y=-11$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$x-y=4$$

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

$$2x+3-y=0$$

Tangohia te 3 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$2x-y=-3$$

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.

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

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

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

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.

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

Tāpiri x ki te -2x.

$$-x=4+3$$

Tāpiri 4 ki te 3.

$$-x=7$$

Whakawehea ngā taha e rua ki te -1.

$$x=-7$$

Whakaurua te -7 mō x ki 2x-y=-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$2\left(-7\right)-y=-3$$

Whakareatia 2 ki te -7.

$$-14-y=-3$$

Me tāpiri 14 ki ngā taha e rua o te whārite.

$$-y=11$$

Whakawehea ngā taha e rua ki te -1.

$$y=-11$$

Kua oti te pūnaha te whakatau.

$$x=-7,y=-11$$