Whakaoti i te {l}{y=-7/3x+3}{y=-2/3x-2}

$$\left. \begin{array} { l } { y = - \frac { 7 } { 3 } x + 3 } \\ { y = - \frac { 2 } { 3 } x - 2 } \end{array} \right.$$

Whakaoti mō y, x

$$x=3$$

$$y=-4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te \frac{7}{3}x ki ngā taha e rua.

$$y+\frac{7}{3}x=3$$

Whakaarohia te whārite tuarua. Me tāpiri te \frac{2}{3}x ki ngā taha e rua.

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

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.

$$y+\frac{7}{3}x=3,y+\frac{2}{3}x=-2$$

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

$$y+\frac{7}{3}x=3$$

Me tango \frac{7x}{3} mai i ngā taha e rua o te whārite.

$$y=-\frac{7}{3}x+3$$

Whakakapia te -\frac{7x}{3}+3 mō te y ki tērā atu whārite, y+\frac{2}{3}x=-2.

$$-\frac{7}{3}x+3+\frac{2}{3}x=-2$$

Tāpiri -\frac{7x}{3} ki te \frac{2x}{3}.

$$-\frac{5}{3}x+3=-2$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$-\frac{5}{3}x=-5$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{5}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=3$$

Whakaurua te 3 mō x ki y=-\frac{7}{3}x+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-\frac{7}{3}\times 3+3$$

Whakareatia -\frac{7}{3} ki te 3.

$$y=-7+3$$

Tāpiri 3 ki te -7.

$$y=-4$$

Kua oti te pūnaha te whakatau.

$$y=-4,x=3$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te \frac{7}{3}x ki ngā taha e rua.

$$y+\frac{7}{3}x=3$$

Whakaarohia te whārite tuarua. Me tāpiri te \frac{2}{3}x ki ngā taha e rua.

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

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

$$y+\frac{7}{3}x=3,y+\frac{2}{3}x=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&\frac{7}{3}\\1&\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}3\\-2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&\frac{7}{3}\\1&\frac{2}{3}\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&\frac{7}{3}\\1&\frac{2}{3}\end{matrix}\right))\left(\begin{matrix}1&\frac{7}{3}\\1&\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&\frac{7}{3}\\1&\frac{2}{3}\end{matrix}\right))\left(\begin{matrix}3\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&\frac{7}{3}\\1&\frac{2}{3}\end{matrix}\right))\left(\begin{matrix}3\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&\frac{7}{3}\\1&\frac{2}{3}\end{matrix}\right))\left(\begin{matrix}3\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{2}{3}}{\frac{2}{3}-\frac{7}{3}}&-\frac{\frac{7}{3}}{\frac{2}{3}-\frac{7}{3}}\\-\frac{1}{\frac{2}{3}-\frac{7}{3}}&\frac{1}{\frac{2}{3}-\frac{7}{3}}\end{matrix}\right)\left(\begin{matrix}3\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{5}&\frac{7}{5}\\\frac{3}{5}&-\frac{3}{5}\end{matrix}\right)\left(\begin{matrix}3\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{5}\times 3+\frac{7}{5}\left(-2\right)\\\frac{3}{5}\times 3-\frac{3}{5}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=-4,x=3$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te \frac{7}{3}x ki ngā taha e rua.

$$y+\frac{7}{3}x=3$$

Whakaarohia te whārite tuarua. Me tāpiri te \frac{2}{3}x ki ngā taha e rua.

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

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.

$$y+\frac{7}{3}x=3,y+\frac{2}{3}x=-2$$

Me tango y+\frac{2}{3}x=-2 mai i y+\frac{7}{3}x=3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y+\frac{7}{3}x-\frac{2}{3}x=3+2$$

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.

$$\frac{7}{3}x-\frac{2}{3}x=3+2$$

Tāpiri \frac{7x}{3} ki te -\frac{2x}{3}.

$$\frac{5}{3}x=3+2$$

Tāpiri 3 ki te 2.

$$\frac{5}{3}x=5$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=3$$

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

$$y+\frac{2}{3}\times 3=-2$$

Whakareatia \frac{2}{3} ki te 3.

$$y+2=-2$$

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

$$y=-4$$

Kua oti te pūnaha te whakatau.

$$y=-4,x=3$$