Whakaoti i te {l}{2x+y=18}{3x+2y=28}

$$\left. \begin{array} { l } { 2 x + y = 18 } \\ { 3 x + 2 y = 28 } \end{array} \right.$$

Whakaoti mō x, y

$$x=8$$

$$y=2$$

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.

$$2x+y=18,3x+2y=28$$

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

$$2x+y=18$$

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

$$2x=-y+18$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-y+18\right)$$

Whakareatia \frac{1}{2} ki te -y+18.

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

Whakakapia te -\frac{y}{2}+9 mō te x ki tērā atu whārite, 3x+2y=28.

$$3\left(-\frac{1}{2}y+9\right)+2y=28$$

Whakareatia 3 ki te -\frac{y}{2}+9.

$$-\frac{3}{2}y+27+2y=28$$

Tāpiri -\frac{3y}{2} ki te 2y.

$$\frac{1}{2}y+27=28$$

Me tango 27 mai i ngā taha e rua o te whārite.

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

Me whakarea ngā taha e rua ki te 2.

$$y=2$$

Whakaurua te 2 mō y ki x=-\frac{1}{2}y+9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times 2+9$$

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

$$x=-1+9$$

Tāpiri 9 ki te -1.

$$x=8$$

Kua oti te pūnaha te whakatau.

$$x=8,y=2$$

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.

$$2x+y=18,3x+2y=28$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&1\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}18\\28\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&1\\3&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&1\\3&2\end{matrix}\right))\left(\begin{matrix}2&1\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\3&2\end{matrix}\right))\left(\begin{matrix}18\\28\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\3&2\end{matrix}\right))\left(\begin{matrix}18\\28\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\3&2\end{matrix}\right))\left(\begin{matrix}18\\28\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2\times 2-3}&-\frac{1}{2\times 2-3}\\-\frac{3}{2\times 2-3}&\frac{2}{2\times 2-3}\end{matrix}\right)\left(\begin{matrix}18\\28\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2&-1\\-3&2\end{matrix}\right)\left(\begin{matrix}18\\28\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\times 18-28\\-3\times 18+2\times 28\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=8,y=2$$

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.

$$2x+y=18,3x+2y=28$$

Kia ōrite ai a 2x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$3\times 2x+3y=3\times 18,2\times 3x+2\times 2y=2\times 28$$

Whakarūnātia.

$$6x+3y=54,6x+4y=56$$

Me tango 6x+4y=56 mai i 6x+3y=54 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+3y-4y=54-56$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3y-4y=54-56$$

Tāpiri 3y ki te -4y.

$$-y=54-56$$

Tāpiri 54 ki te -56.

$$-y=-2$$

Whakawehea ngā taha e rua ki te -1.

$$y=2$$

Whakaurua te 2 mō y ki 3x+2y=28. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+2\times 2=28$$

Whakareatia 2 ki te 2.

$$3x+4=28$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$3x=24$$

Whakawehea ngā taha e rua ki te 3.

$$x=8$$

Kua oti te pūnaha te whakatau.

$$x=8,y=2$$