Whakaoti i te {c}{y=x+6}{y=1/2x+4}

$$\left. \begin{array} { c } { y = x + 6 } \\ { y = \frac { 1 } { 2 } x + 4 } \end{array} \right.$$

Whakaoti mō y, x

$$x=-4$$

$$y=2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-x=6$$

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

$$y-\frac{1}{2}x=4$$

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.

$$y-x=6,y-\frac{1}{2}x=4$$

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

$$y-x=6$$

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

$$y=x+6$$

Whakakapia te x+6 mō te y ki tērā atu whārite, y-\frac{1}{2}x=4.

$$x+6-\frac{1}{2}x=4$$

Tāpiri x ki te -\frac{x}{2}.

$$\frac{1}{2}x+6=4$$

Me tango 6 mai i ngā taha e rua o te whārite.

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

Me whakarea ngā taha e rua ki te 2.

$$x=-4$$

Whakaurua te -4 mō x ki y=x+6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-4+6$$

Tāpiri 6 ki te -4.

$$y=2$$

Kua oti te pūnaha te whakatau.

$$y=2,x=-4$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-x=6$$

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

$$y-\frac{1}{2}x=4$$

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

$$y-x=6,y-\frac{1}{2}x=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\1&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}6\\4\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-1\\1&-\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}1&-1\\1&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&-\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}6\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&-\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}6\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&-\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}6\\4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{\frac{1}{2}}{-\frac{1}{2}-\left(-1\right)}&-\frac{-1}{-\frac{1}{2}-\left(-1\right)}\\-\frac{1}{-\frac{1}{2}-\left(-1\right)}&\frac{1}{-\frac{1}{2}-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}6\\4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-1&2\\-2&2\end{matrix}\right)\left(\begin{matrix}6\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-6+2\times 4\\-2\times 6+2\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}2\\-4\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=2,x=-4$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-x=6$$

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

$$y-\frac{1}{2}x=4$$

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.

$$y-x=6,y-\frac{1}{2}x=4$$

Me tango y-\frac{1}{2}x=4 mai i y-x=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-x+\frac{1}{2}x=6-4$$

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.

$$-x+\frac{1}{2}x=6-4$$

Tāpiri -x ki te \frac{x}{2}.

$$-\frac{1}{2}x=6-4$$

Tāpiri 6 ki te -4.

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

Me whakarea ngā taha e rua ki te -2.

$$x=-4$$

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

$$y-\frac{1}{2}\left(-4\right)=4$$

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

$$y+2=4$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$y=2$$

Kua oti te pūnaha te whakatau.

$$y=2,x=-4$$