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

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

Whakaoti mō y, x

$$x=2$$

$$y=3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y+x=5$$

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

$$y-2x=-1$$

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.

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

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

$$y+x=5$$

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

$$y=-x+5$$

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

$$-x+5-2x=-1$$

Tāpiri -x ki te -2x.

$$-3x+5=-1$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$-3x=-6$$

Whakawehea ngā taha e rua ki te -3.

$$x=2$$

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

$$y=-2+5$$

Tāpiri 5 ki te -2.

$$y=3$$

Kua oti te pūnaha te whakatau.

$$y=3,x=2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y+x=5$$

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

$$y-2x=-1$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}&\frac{1}{3}\\\frac{1}{3}&-\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}5\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}\times 5+\frac{1}{3}\left(-1\right)\\\frac{1}{3}\times 5-\frac{1}{3}\left(-1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=3,x=2$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y+x=5$$

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

$$y-2x=-1$$

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.

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

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

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

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.

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

Tāpiri x ki te 2x.

$$3x=5+1$$

Tāpiri 5 ki te 1.

$$3x=6$$

Whakawehea ngā taha e rua ki te 3.

$$x=2$$

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.

$$y-4=-1$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$y=3$$

Kua oti te pūnaha te whakatau.

$$y=3,x=2$$