Whakaoti i te {l}{x/4-y/4=-1}{x+4y=-9}

$$\left. \begin{array} { l } { \frac { x } { 4 } - \frac { y } { 4 } = - 1 } \\ { x + 4 y = - 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-5$$

$$y=-1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 4.

$$x-y=-4$$

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.

$$x-y=-4,x+4y=-9$$

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

$$x-y=-4$$

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

$$x=y-4$$

Whakakapia te y-4 mō te x ki tērā atu whārite, x+4y=-9.

$$y-4+4y=-9$$

Tāpiri y ki te 4y.

$$5y-4=-9$$

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

$$5y=-5$$

Whakawehea ngā taha e rua ki te 5.

$$y=-1$$

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

$$x=-1-4$$

Tāpiri -4 ki te -1.

$$x=-5$$

Kua oti te pūnaha te whakatau.

$$x=-5,y=-1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 4.

$$x-y=-4$$

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

$$x-y=-4,x+4y=-9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\1&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\\-9\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-1\\1&4\end{matrix}\right))\left(\begin{matrix}1&-1\\1&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&4\end{matrix}\right))\left(\begin{matrix}-4\\-9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&4\end{matrix}\right))\left(\begin{matrix}-4\\-9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\1&4\end{matrix}\right))\left(\begin{matrix}-4\\-9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{4-\left(-1\right)}&-\frac{-1}{4-\left(-1\right)}\\-\frac{1}{4-\left(-1\right)}&\frac{1}{4-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}-4\\-9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{5}&\frac{1}{5}\\-\frac{1}{5}&\frac{1}{5}\end{matrix}\right)\left(\begin{matrix}-4\\-9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{5}\left(-4\right)+\frac{1}{5}\left(-9\right)\\-\frac{1}{5}\left(-4\right)+\frac{1}{5}\left(-9\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\\-1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-5,y=-1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 4.

$$x-y=-4$$

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.

$$x-y=-4,x+4y=-9$$

Me tango x+4y=-9 mai i x-y=-4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x-y-4y=-4+9$$

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.

$$-y-4y=-4+9$$

Tāpiri -y ki te -4y.

$$-5y=-4+9$$

Tāpiri -4 ki te 9.

$$-5y=5$$

Whakawehea ngā taha e rua ki te -5.

$$y=-1$$

Whakaurua te -1 mō y ki x+4y=-9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+4\left(-1\right)=-9$$

Whakareatia 4 ki te -1.

$$x-4=-9$$

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

$$x=-5$$

Kua oti te pūnaha te whakatau.

$$x=-5,y=-1$$