Whakaoti i te {l}{2x-y=4}{y=x+3}

$$\left. \begin{array}{l}{ 2 x - y = 4 }\\{ y = x + 3 }\end{array} \right.$$

Whakaoti mō x, y

$$x=7$$

$$y=10$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-x=3$$

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.

$$2x-y=4,-x+y=3$$

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

$$2x-y=4$$

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

$$2x=y+4$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(y+4\right)$$

Whakareatia \frac{1}{2} ki te y+4.

$$x=\frac{1}{2}y+2$$

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

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

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

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

Tāpiri -\frac{y}{2} ki te y.

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

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$\frac{1}{2}y=5$$

Me whakarea ngā taha e rua ki te 2.

$$y=10$$

Whakaurua te 10 mō y ki x=\frac{1}{2}y+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{2}\times 10+2$$

Whakareatia \frac{1}{2} ki te 10.

$$x=5+2$$

Tāpiri 2 ki te 5.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=10$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-x=3$$

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

$$2x-y=4,-x+y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-1\\-1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\3\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&-1\\-1&1\end{matrix}\right))\left(\begin{matrix}2&-1\\-1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\-1&1\end{matrix}\right))\left(\begin{matrix}4\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\-1&1\end{matrix}\right))\left(\begin{matrix}4\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\-1&1\end{matrix}\right))\left(\begin{matrix}4\\3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2-\left(-\left(-1\right)\right)}&-\frac{-1}{2-\left(-\left(-1\right)\right)}\\-\frac{-1}{2-\left(-\left(-1\right)\right)}&\frac{2}{2-\left(-\left(-1\right)\right)}\end{matrix}\right)\left(\begin{matrix}4\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1&1\\1&2\end{matrix}\right)\left(\begin{matrix}4\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4+3\\4+2\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\10\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=7,y=10$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-x=3$$

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.

$$2x-y=4,-x+y=3$$

Kia ōrite ai a 2x me -x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$-2x-\left(-y\right)=-4,2\left(-1\right)x+2y=2\times 3$$

Whakarūnātia.

$$-2x+y=-4,-2x+2y=6$$

Me tango -2x+2y=6 mai i -2x+y=-4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x+y-2y=-4-6$$

Tāpiri -2x ki te 2x. Ka whakakore atu ngā kupu -2x me 2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$y-2y=-4-6$$

Tāpiri y ki te -2y.

$$-y=-4-6$$

Tāpiri -4 ki te -6.

$$-y=-10$$

Whakawehea ngā taha e rua ki te -1.

$$y=10$$

Whakaurua te 10 mō y ki -x+y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-x+10=3$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$-x=-7$$

Whakawehea ngā taha e rua ki te -1.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=10$$