Whakaoti i te {l}{y=-1/2x+1}{y=1/2x-3}

$$\left. \begin{array} { l } { y = - \frac { 1 } { 2 } x + 1 } \\ { y = \frac { 1 } { 2 } x - 3 } \end{array} \right.$$

Whakaoti mō y, x

$$x=4$$

$$y=-1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te \frac{1}{2}x ki ngā taha e rua.

$$y+\frac{1}{2}x=1$$

Whakaarohia te whārite tuarua. Tangohia te \frac{1}{2}x mai i ngā taha e rua.

$$y-\frac{1}{2}x=-3$$

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.

$$y+\frac{1}{2}x=1,y-\frac{1}{2}x=-3$$

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

$$y+\frac{1}{2}x=1$$

Me tango \frac{x}{2} mai i ngā taha e rua o te whārite.

$$y=-\frac{1}{2}x+1$$

Whakakapia te -\frac{x}{2}+1 mō te y ki tērā atu whārite, y-\frac{1}{2}x=-3.

$$-\frac{1}{2}x+1-\frac{1}{2}x=-3$$

Tāpiri -\frac{x}{2} ki te -\frac{x}{2}.

$$-x+1=-3$$

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

$$-x=-4$$

Whakawehea ngā taha e rua ki te -1.

$$x=4$$

Whakaurua te 4 mō x ki y=-\frac{1}{2}x+1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-\frac{1}{2}\times 4+1$$

Whakareatia -\frac{1}{2} ki te 4.

$$y=-2+1$$

Tāpiri 1 ki te -2.

$$y=-1$$

Kua oti te pūnaha te whakatau.

$$y=-1,x=4$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te \frac{1}{2}x ki ngā taha e rua.

$$y+\frac{1}{2}x=1$$

Whakaarohia te whārite tuarua. Tangohia te \frac{1}{2}x mai i ngā taha e rua.

$$y-\frac{1}{2}x=-3$$

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

$$y+\frac{1}{2}x=1,y-\frac{1}{2}x=-3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=-1,x=4$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te \frac{1}{2}x ki ngā taha e rua.

$$y+\frac{1}{2}x=1$$

Whakaarohia te whārite tuarua. Tangohia te \frac{1}{2}x mai i ngā taha e rua.

$$y-\frac{1}{2}x=-3$$

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.

$$y+\frac{1}{2}x=1,y-\frac{1}{2}x=-3$$

Me tango y-\frac{1}{2}x=-3 mai i y+\frac{1}{2}x=1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y+\frac{1}{2}x+\frac{1}{2}x=1+3$$

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.

$$\frac{1}{2}x+\frac{1}{2}x=1+3$$

Tāpiri \frac{x}{2} ki te \frac{x}{2}.

$$x=1+3$$

Tāpiri 1 ki te 3.

$$x=4$$

Whakaurua te 4 mō x ki y-\frac{1}{2}x=-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y-\frac{1}{2}\times 4=-3$$

Whakareatia -\frac{1}{2} ki te 4.

$$y-2=-3$$

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

$$y=-1$$

Kua oti te pūnaha te whakatau.

$$y=-1,x=4$$