Whakaoti i te {l}{x+y=40}{y+2=2x}

$$\left\{ \begin{array} { l } { x + y = 40 } \\ { y + 2 = 2 x } \end{array} \right.$$

Whakaoti mō x, y

$$x=14$$

$$y=26$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

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

Tangohia te 2 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$y-2x=-2$$

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.

$$x+y=40,-2x+y=-2$$

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

$$x+y=40$$

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

$$x=-y+40$$

Whakakapia te -y+40 mō te x ki tērā atu whārite, -2x+y=-2.

$$-2\left(-y+40\right)+y=-2$$

Whakareatia -2 ki te -y+40.

$$2y-80+y=-2$$

Tāpiri 2y ki te y.

$$3y-80=-2$$

Me tāpiri 80 ki ngā taha e rua o te whārite.

$$3y=78$$

Whakawehea ngā taha e rua ki te 3.

$$y=26$$

Whakaurua te 26 mō y ki x=-y+40. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-26+40$$

Tāpiri 40 ki te -26.

$$x=14$$

Kua oti te pūnaha te whakatau.

$$x=14,y=26$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

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

Tangohia te 2 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$y-2x=-2$$

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

$$x+y=40,-2x+y=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\-2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}40\\-2\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\-2&1\end{matrix}\right))\left(\begin{matrix}1&1\\-2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-2&1\end{matrix}\right))\left(\begin{matrix}40\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-2&1\end{matrix}\right))\left(\begin{matrix}40\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-2&1\end{matrix}\right))\left(\begin{matrix}40\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-\left(-2\right)}&-\frac{1}{1-\left(-2\right)}\\-\frac{-2}{1-\left(-2\right)}&\frac{1}{1-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}40\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&-\frac{1}{3}\\\frac{2}{3}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}40\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 40-\frac{1}{3}\left(-2\right)\\\frac{2}{3}\times 40+\frac{1}{3}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14\\26\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=14,y=26$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

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

Tangohia te 2 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$y-2x=-2$$

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.

$$x+y=40,-2x+y=-2$$

Me tango -2x+y=-2 mai i x+y=40 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x+2x+y-y=40+2$$

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.

$$x+2x=40+2$$

Tāpiri x ki te 2x.

$$3x=40+2$$

Tāpiri 40 ki te 2.

$$3x=42$$

Whakawehea ngā taha e rua ki te 3.

$$x=14$$

Whakaurua te 14 mō x ki -2x+y=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-2\times 14+y=-2$$

Whakareatia -2 ki te 14.

$$-28+y=-2$$

Me tāpiri 28 ki ngā taha e rua o te whārite.

$$y=26$$

Kua oti te pūnaha te whakatau.

$$x=14,y=26$$