Whakaoti i te {c}{4x-y=18}{3x+5y=2}

$$\left\{ \begin{array} { c } { 4 x - y = 18 } \\ { 3 x + 5 y = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=-2$$

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.

$$4x-y=18,3x+5y=2$$

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

$$4x-y=18$$

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

$$4x=y+18$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(y+18\right)$$

Whakareatia \frac{1}{4} ki te y+18.

$$x=\frac{1}{4}y+\frac{9}{2}$$

Whakakapia te \frac{y}{4}+\frac{9}{2} mō te x ki tērā atu whārite, 3x+5y=2.

$$3\left(\frac{1}{4}y+\frac{9}{2}\right)+5y=2$$

Whakareatia 3 ki te \frac{y}{4}+\frac{9}{2}.

$$\frac{3}{4}y+\frac{27}{2}+5y=2$$

Tāpiri \frac{3y}{4} ki te 5y.

$$\frac{23}{4}y+\frac{27}{2}=2$$

Me tango \frac{27}{2} mai i ngā taha e rua o te whārite.

$$\frac{23}{4}y=-\frac{23}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{23}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-2$$

Whakaurua te -2 mō y ki x=\frac{1}{4}y+\frac{9}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

Whakareatia \frac{1}{4} ki te -2.

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

Tāpiri \frac{9}{2} ki te -\frac{1}{2} 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.

$$x=4$$

Kua oti te pūnaha te whakatau.

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

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.

$$4x-y=18,3x+5y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-1\\3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}18\\2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-1\\3&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&-1\\3&5\end{matrix}\right))\left(\begin{matrix}4&-1\\3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\3&5\end{matrix}\right))\left(\begin{matrix}18\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\3&5\end{matrix}\right))\left(\begin{matrix}18\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\3&5\end{matrix}\right))\left(\begin{matrix}18\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{4\times 5-\left(-3\right)}&-\frac{-1}{4\times 5-\left(-3\right)}\\-\frac{3}{4\times 5-\left(-3\right)}&\frac{4}{4\times 5-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}18\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{23}&\frac{1}{23}\\-\frac{3}{23}&\frac{4}{23}\end{matrix}\right)\left(\begin{matrix}18\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{23}\times 18+\frac{1}{23}\times 2\\-\frac{3}{23}\times 18+\frac{4}{23}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$4x-y=18,3x+5y=2$$

Kia ōrite ai a 4x 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 4.

$$3\times 4x+3\left(-1\right)y=3\times 18,4\times 3x+4\times 5y=4\times 2$$

Whakarūnātia.

$$12x-3y=54,12x+20y=8$$

Me tango 12x+20y=8 mai i 12x-3y=54 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x-3y-20y=54-8$$

Tāpiri 12x ki te -12x. Ka whakakore atu ngā kupu 12x me -12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-3y-20y=54-8$$

Tāpiri -3y ki te -20y.

$$-23y=54-8$$

Tāpiri 54 ki te -8.

$$-23y=46$$

Whakawehea ngā taha e rua ki te -23.

$$y=-2$$

Whakaurua te -2 mō y ki 3x+5y=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+5\left(-2\right)=2$$

Whakareatia 5 ki te -2.

$$3x-10=2$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$3x=12$$

Whakawehea ngā taha e rua ki te 3.

$$x=4$$

Kua oti te pūnaha te whakatau.

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