Whakaoti i te {c}{y=2x+1}{y=-5x+15}

$$\left\{ \begin{array} { c } { y = 2 x + 1 } \\ { y = - 5 x + 15 } \end{array} \right.$$

Whakaoti mō y, x

$$x=2$$

$$y=5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-2x=1$$

Whakaarohia te whārite tuarua. Me tāpiri te 5x ki ngā taha e rua.

$$y+5x=15$$

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.

$$y-2x=1,y+5x=15$$

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

$$y-2x=1$$

Me tāpiri 2x ki ngā taha e rua o te whārite.

$$y=2x+1$$

Whakakapia te 2x+1 mō te y ki tērā atu whārite, y+5x=15.

$$2x+1+5x=15$$

Tāpiri 2x ki te 5x.

$$7x+1=15$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$7x=14$$

Whakawehea ngā taha e rua ki te 7.

$$x=2$$

Whakaurua te 2 mō x ki y=2x+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=2\times 2+1$$

Whakareatia 2 ki te 2.

$$y=4+1$$

Tāpiri 1 ki te 4.

$$y=5$$

Kua oti te pūnaha te whakatau.

$$y=5,x=2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-2x=1$$

Whakaarohia te whārite tuarua. Me tāpiri te 5x ki ngā taha e rua.

$$y+5x=15$$

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

$$y-2x=1,y+5x=15$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-2\\1&5\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}1\\15\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-2\\1&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-2\\1&5\end{matrix}\right))\left(\begin{matrix}1&-2\\1&5\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\1&5\end{matrix}\right))\left(\begin{matrix}1\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\1&5\end{matrix}\right))\left(\begin{matrix}1\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2\\1&5\end{matrix}\right))\left(\begin{matrix}1\\15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5-\left(-2\right)}&-\frac{-2}{5-\left(-2\right)}\\-\frac{1}{5-\left(-2\right)}&\frac{1}{5-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}1\\15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{5}{7}&\frac{2}{7}\\-\frac{1}{7}&\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}1\\15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{5}{7}+\frac{2}{7}\times 15\\-\frac{1}{7}+\frac{1}{7}\times 15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}5\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=5,x=2$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-2x=1$$

Whakaarohia te whārite tuarua. Me tāpiri te 5x ki ngā taha e rua.

$$y+5x=15$$

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.

$$y-2x=1,y+5x=15$$

Me tango y+5x=15 mai i y-2x=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-2x-5x=1-15$$

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.

$$-2x-5x=1-15$$

Tāpiri -2x ki te -5x.

$$-7x=1-15$$

Tāpiri 1 ki te -15.

$$-7x=-14$$

Whakawehea ngā taha e rua ki te -7.

$$x=2$$

Whakaurua te 2 mō x ki y+5x=15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y+5\times 2=15$$

Whakareatia 5 ki te 2.

$$y+10=15$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$y=5$$

Kua oti te pūnaha te whakatau.

$$y=5,x=2$$