Whakaoti i te {l}{x-y=13}{x=6y-7}

$$\left\{ \begin{array} { l } { x - y = 13 } \\ { x = 6 y - 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x=17$$

$$y=4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 6y mai i ngā taha e rua.

$$x-6y=-7$$

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.

$$x-y=13,x-6y=-7$$

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

$$x-y=13$$

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

$$x=y+13$$

Whakakapia te y+13 mō te x ki tērā atu whārite, x-6y=-7.

$$y+13-6y=-7$$

Tāpiri y ki te -6y.

$$-5y+13=-7$$

Me tango 13 mai i ngā taha e rua o te whārite.

$$-5y=-20$$

Whakawehea ngā taha e rua ki te -5.

$$y=4$$

Whakaurua te 4 mō y ki x=y+13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=4+13$$

Tāpiri 13 ki te 4.

$$x=17$$

Kua oti te pūnaha te whakatau.

$$x=17,y=4$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 6y mai i ngā taha e rua.

$$x-6y=-7$$

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

$$x-y=13,x-6y=-7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\1&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}13\\-7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-1\\1&-6\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-1\\1&-6\end{matrix}\right))\left(\begin{matrix}1&-1\\1&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&-6\end{matrix}\right))\left(\begin{matrix}13\\-7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&-6\end{matrix}\right))\left(\begin{matrix}13\\-7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&-6\end{matrix}\right))\left(\begin{matrix}13\\-7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{-6-\left(-1\right)}&-\frac{-1}{-6-\left(-1\right)}\\-\frac{1}{-6-\left(-1\right)}&\frac{1}{-6-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}13\\-7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{5}&-\frac{1}{5}\\\frac{1}{5}&-\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}13\\-7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{5}\times 13-\frac{1}{5}\left(-7\right)\\\frac{1}{5}\times 13-\frac{1}{5}\left(-7\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}17\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=17,y=4$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 6y mai i ngā taha e rua.

$$x-6y=-7$$

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.

$$x-y=13,x-6y=-7$$

Me tango x-6y=-7 mai i x-y=13 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x-y+6y=13+7$$

Tāpiri x ki te -x. Ka whakakore atu ngā kupu x me -x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-y+6y=13+7$$

Tāpiri -y ki te 6y.

$$5y=13+7$$

Tāpiri 13 ki te 7.

$$5y=20$$

Whakawehea ngā taha e rua ki te 5.

$$y=4$$

Whakaurua te 4 mō y ki x-6y=-7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-6\times 4=-7$$

Whakareatia -6 ki te 4.

$$x-24=-7$$

Me tāpiri 24 ki ngā taha e rua o te whārite.

$$x=17$$

Kua oti te pūnaha te whakatau.

$$x=17,y=4$$