Whakaoti i te {l}{y=x-18}{y=15x}

$$\left\{ \begin{array} { l } { y = x - 18 } \\ { y = 15 x } \end{array} \right.$$

Whakaoti mō y, x

$$x = -\frac{9}{7} = -1\frac{2}{7} \approx -1.285714286$$

$$y = -\frac{135}{7} = -19\frac{2}{7} \approx -19.285714286$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-x=-18$$

Whakaarohia te whārite tuarua. Tangohia te 15x mai i ngā taha e rua.

$$y-15x=0$$

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.

$$y-x=-18,y-15x=0$$

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

$$y-x=-18$$

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

$$y=x-18$$

Whakakapia te x-18 mō te y ki tērā atu whārite, y-15x=0.

$$x-18-15x=0$$

Tāpiri x ki te -15x.

$$-14x-18=0$$

Me tāpiri 18 ki ngā taha e rua o te whārite.

$$-14x=18$$

Whakawehea ngā taha e rua ki te -14.

$$x=-\frac{9}{7}$$

Whakaurua te -\frac{9}{7} mō x ki y=x-18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-\frac{9}{7}-18$$

Tāpiri -18 ki te -\frac{9}{7}.

$$y=-\frac{135}{7}$$

Kua oti te pūnaha te whakatau.

$$y=-\frac{135}{7},x=-\frac{9}{7}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-x=-18$$

Whakaarohia te whārite tuarua. Tangohia te 15x mai i ngā taha e rua.

$$y-15x=0$$

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

$$y-x=-18,y-15x=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\1&-15\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-18\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-1\\1&-15\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-1\\1&-15\end{matrix}\right))\left(\begin{matrix}1&-1\\1&-15\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&-15\end{matrix}\right))\left(\begin{matrix}-18\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&-15\end{matrix}\right))\left(\begin{matrix}-18\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&-15\end{matrix}\right))\left(\begin{matrix}-18\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{15}{-15-\left(-1\right)}&-\frac{-1}{-15-\left(-1\right)}\\-\frac{1}{-15-\left(-1\right)}&\frac{1}{-15-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}-18\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{15}{14}&-\frac{1}{14}\\\frac{1}{14}&-\frac{1}{14}\end{matrix}\right)\left(\begin{matrix}-18\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{15}{14}\left(-18\right)\\\frac{1}{14}\left(-18\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{135}{7}\\-\frac{9}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=-\frac{135}{7},x=-\frac{9}{7}$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-x=-18$$

Whakaarohia te whārite tuarua. Tangohia te 15x mai i ngā taha e rua.

$$y-15x=0$$

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.

$$y-x=-18,y-15x=0$$

Me tango y-15x=0 mai i y-x=-18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-x+15x=-18$$

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.

$$-x+15x=-18$$

Tāpiri -x ki te 15x.

$$14x=-18$$

Whakawehea ngā taha e rua ki te 14.

$$x=-\frac{9}{7}$$

Whakaurua te -\frac{9}{7} mō x ki y-15x=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y-15\left(-\frac{9}{7}\right)=0$$

Whakareatia -15 ki te -\frac{9}{7}.

$$y+\frac{135}{7}=0$$

Me tango \frac{135}{7} mai i ngā taha e rua o te whārite.

$$y=-\frac{135}{7}$$

Kua oti te pūnaha te whakatau.

$$y=-\frac{135}{7},x=-\frac{9}{7}$$