Whakaoti i te {l}{4x+y=9}{2x+y=7}

$$\left\{ \begin{array} { l } { 4 x + y = 9 } \\ { 2 x + y = 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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.

$$4x+y=9,2x+y=7$$

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

$$4x+y=9$$

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

$$4x=-y+9$$

Whakawehea ngā taha e rua ki te 4.

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

Whakareatia \frac{1}{4} ki te -y+9.

$$x=-\frac{1}{4}y+\frac{9}{4}$$

Whakakapia te \frac{-y+9}{4} mō te x ki tērā atu whārite, 2x+y=7.

$$2\left(-\frac{1}{4}y+\frac{9}{4}\right)+y=7$$

Whakareatia 2 ki te \frac{-y+9}{4}.

$$-\frac{1}{2}y+\frac{9}{2}+y=7$$

Tāpiri -\frac{y}{2} ki te y.

$$\frac{1}{2}y+\frac{9}{2}=7$$

Me tango \frac{9}{2} mai i ngā taha e rua o te whārite.

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

Me whakarea ngā taha e rua ki te 2.

$$y=5$$

Whakaurua te 5 mō y ki x=-\frac{1}{4}y+\frac{9}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{4}\times 5+\frac{9}{4}$$

Whakareatia -\frac{1}{4} ki te 5.

$$x=\frac{-5+9}{4}$$

Tāpiri \frac{9}{4} ki te -\frac{5}{4} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=5$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$4x+y=9,2x+y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&1\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&1\\2&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&1\\2&1\end{matrix}\right))\left(\begin{matrix}4&1\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&1\\2&1\end{matrix}\right))\left(\begin{matrix}9\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&1\\2&1\end{matrix}\right))\left(\begin{matrix}9\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&1\\2&1\end{matrix}\right))\left(\begin{matrix}9\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4-2}&-\frac{1}{4-2}\\-\frac{2}{4-2}&\frac{4}{4-2}\end{matrix}\right)\left(\begin{matrix}9\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=5$$

Ngā Upane e Whakamahi i te Tangohanga

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.

$$4x+y=9,2x+y=7$$

Me tango 2x+y=7 mai i 4x+y=9 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-2x+y-y=9-7$$

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.

$$4x-2x=9-7$$

Tāpiri 4x ki te -2x.

$$2x=9-7$$

Tāpiri 9 ki te -7.

$$2x=2$$

Whakawehea ngā taha e rua ki te 2.

$$x=1$$

Whakaurua te 1 mō x ki 2x+y=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$2+y=7$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$y=5$$

Kua oti te pūnaha te whakatau.

$$x=1,y=5$$