Whakaoti i te {l}{x=y+4}{4x-y=22}

$$\left. \begin{array} { l } { x = y + 4 } \\ { 4 x - y = 22 } \end{array} \right.$$

Whakaoti mō x, y

$$x=6$$

$$y=2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$x-y=4$$

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.

$$x-y=4,4x-y=22$$

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, 4x-y=22.

$$4\left(y+4\right)-y=22$$

Whakareatia 4 ki te y+4.

$$4y+16-y=22$$

Tāpiri 4y ki te -y.

$$3y+16=22$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$3y=6$$

Whakawehea ngā taha e rua ki te 3.

$$y=2$$

Whakaurua te 2 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.

$$x=2+4$$

Tāpiri 4 ki te 2.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$x-y=4$$

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

$$x-y=4,4x-y=22$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\22\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-1\\4&-1\end{matrix}\right))\left(\begin{matrix}1&-1\\4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\4&-1\end{matrix}\right))\left(\begin{matrix}4\\22\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\4&-1\end{matrix}\right))\left(\begin{matrix}4\\22\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\4&-1\end{matrix}\right))\left(\begin{matrix}4\\22\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-\left(-4\right)}&-\frac{-1}{-1-\left(-4\right)}\\-\frac{4}{-1-\left(-4\right)}&\frac{1}{-1-\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}4\\22\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&\frac{1}{3}\\-\frac{4}{3}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}4\\22\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}\times 4+\frac{1}{3}\times 22\\-\frac{4}{3}\times 4+\frac{1}{3}\times 22\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=6,y=2$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$x-y=4$$

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.

$$x-y=4,4x-y=22$$

Me tango 4x-y=22 mai i x-y=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-4x-y+y=4-22$$

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.

$$x-4x=4-22$$

Tāpiri x ki te -4x.

$$-3x=4-22$$

Tāpiri 4 ki te -22.

$$-3x=-18$$

Whakawehea ngā taha e rua ki te -3.

$$x=6$$

Whakaurua te 6 mō x ki 4x-y=22. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$4\times 6-y=22$$

Whakareatia 4 ki te 6.

$$24-y=22$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$-y=-2$$

Whakawehea ngā taha e rua ki te -1.

$$y=2$$

Kua oti te pūnaha te whakatau.

$$x=6,y=2$$