Whakaoti i te {l}{2x+y=-6}{y=7x+3}

$$\left. \begin{array} { l } { 2 x + y = - 6 } \\ { y = 7 x + 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-1$$

$$y=-4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 7x mai i ngā taha e rua.

$$y-7x=3$$

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.

$$2x+y=-6,-7x+y=3$$

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

$$2x+y=-6$$

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

$$2x=-y-6$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-y-6\right)$$

Whakareatia \frac{1}{2} ki te -y-6.

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

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

$$-7\left(-\frac{1}{2}y-3\right)+y=3$$

Whakareatia -7 ki te -\frac{y}{2}-3.

$$\frac{7}{2}y+21+y=3$$

Tāpiri \frac{7y}{2} ki te y.

$$\frac{9}{2}y+21=3$$

Me tango 21 mai i ngā taha e rua o te whārite.

$$\frac{9}{2}y=-18$$

Whakawehea ngā taha e rua o te whārite ki te \frac{9}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-4$$

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

$$x=-\frac{1}{2}\left(-4\right)-3$$

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

$$x=2-3$$

Tāpiri -3 ki te 2.

$$x=-1$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 7x mai i ngā taha e rua.

$$y-7x=3$$

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

$$2x+y=-6,-7x+y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&1\\-7&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-6\\3\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&1\\-7&1\end{matrix}\right))\left(\begin{matrix}2&1\\-7&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\-7&1\end{matrix}\right))\left(\begin{matrix}-6\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\-7&1\end{matrix}\right))\left(\begin{matrix}-6\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\-7&1\end{matrix}\right))\left(\begin{matrix}-6\\3\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2-\left(-7\right)}&-\frac{1}{2-\left(-7\right)}\\-\frac{-7}{2-\left(-7\right)}&\frac{2}{2-\left(-7\right)}\end{matrix}\right)\left(\begin{matrix}-6\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}&-\frac{1}{9}\\\frac{7}{9}&\frac{2}{9}\end{matrix}\right)\left(\begin{matrix}-6\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}\left(-6\right)-\frac{1}{9}\times 3\\\frac{7}{9}\left(-6\right)+\frac{2}{9}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 7x mai i ngā taha e rua.

$$y-7x=3$$

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.

$$2x+y=-6,-7x+y=3$$

Me tango -7x+y=3 mai i 2x+y=-6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x+7x+y-y=-6-3$$

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.

$$2x+7x=-6-3$$

Tāpiri 2x ki te 7x.

$$9x=-6-3$$

Tāpiri -6 ki te -3.

$$9x=-9$$

Whakawehea ngā taha e rua ki te 9.

$$x=-1$$

Whakaurua te -1 mō x ki -7x+y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-7\left(-1\right)+y=3$$

Whakareatia -7 ki te -1.

$$7+y=3$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$y=-4$$

Kua oti te pūnaha te whakatau.

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