Whakaoti i te {l}{y=1/2x-4}{y=-1/4x-1}

$$\left. \begin{array} { l } { y = \frac { 1 } { 2 } x - 4 } \\ { y = - \frac { 1 } { 4 } x - 1 } \end{array} \right.$$

Whakaoti mō y, x

$$x=4$$

$$y=-2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

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

Whakaarohia te whārite tuarua. Me tāpiri te \frac{1}{4}x ki ngā taha e rua.

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

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.

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

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

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

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

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

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

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

Tāpiri \frac{x}{2} ki te \frac{x}{4}.

$$\frac{3}{4}x-4=-1$$

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

$$\frac{3}{4}x=3$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=4$$

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.

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

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

$$y=2-4$$

Tāpiri -4 ki te 2.

$$y=-2$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

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

Whakaarohia te whārite tuarua. Me tāpiri te \frac{1}{4}x ki ngā taha e rua.

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

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-\frac{1}{2}\\1&\frac{1}{4}\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&\frac{2}{3}\\-\frac{4}{3}&\frac{4}{3}\end{matrix}\right)\left(\begin{matrix}-4\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\left(-4\right)+\frac{2}{3}\left(-1\right)\\-\frac{4}{3}\left(-4\right)+\frac{4}{3}\left(-1\right)\end{matrix}\right)$$

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 \frac{1}{2}x mai i ngā taha e rua.

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

Whakaarohia te whārite tuarua. Me tāpiri te \frac{1}{4}x ki ngā taha e rua.

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

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.

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

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

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

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.

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

Tāpiri -\frac{x}{2} ki te -\frac{x}{4}.

$$-\frac{3}{4}x=-4+1$$

Tāpiri -4 ki te 1.

$$-\frac{3}{4}x=-3$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{3}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=4$$

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

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

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

$$y+1=-1$$

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

$$y=-2$$

Kua oti te pūnaha te whakatau.

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