Whakaoti i te {l}{x+15=y}{4x=y}

$$\left. \begin{array} { l } { x + 15 = y } \\ { 4 x = y } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=20$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$x+15-y=0$$

Tangohia te 15 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x-y=-15$$

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

$$4x-y=0$$

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.

$$x-y=-15,4x-y=0$$

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

$$x-y=-15$$

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

$$x=y-15$$

Whakakapia te y-15 mō te x ki tērā atu whārite, 4x-y=0.

$$4\left(y-15\right)-y=0$$

Whakareatia 4 ki te y-15.

$$4y-60-y=0$$

Tāpiri 4y ki te -y.

$$3y-60=0$$

Me tāpiri 60 ki ngā taha e rua o te whārite.

$$3y=60$$

Whakawehea ngā taha e rua ki te 3.

$$y=20$$

Whakaurua te 20 mō y ki x=y-15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=20-15$$

Tāpiri -15 ki te 20.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=20$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$x+15-y=0$$

Tangohia te 15 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x-y=-15$$

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

$$4x-y=0$$

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

$$x-y=-15,4x-y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\20\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=20$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$x+15-y=0$$

Tangohia te 15 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x-y=-15$$

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

$$4x-y=0$$

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.

$$x-y=-15,4x-y=0$$

Me tango 4x-y=0 mai i x-y=-15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

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

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.

$$x-4x=-15$$

Tāpiri x ki te -4x.

$$-3x=-15$$

Whakawehea ngā taha e rua ki te -3.

$$x=5$$

Whakaurua te 5 mō x ki 4x-y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$4\times 5-y=0$$

Whakareatia 4 ki te 5.

$$20-y=0$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$-y=-20$$

Whakawehea ngā taha e rua ki te -1.

$$y=20$$

Kua oti te pūnaha te whakatau.

$$x=5,y=20$$