Whakaoti i te {l}{21x+7y=42}{-5x+5y=10}

$$\left\{ \begin{array} { l } { 21 x + 7 y = 42 } \\ { - 5 x + 5 y = 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=3$$

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.

$$21x+7y=42,-5x+5y=10$$

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

$$21x+7y=42$$

Me tango 7y mai i ngā taha e rua o te whārite.

$$21x=-7y+42$$

Whakawehea ngā taha e rua ki te 21.

$$x=\frac{1}{21}\left(-7y+42\right)$$

Whakareatia \frac{1}{21} ki te -7y+42.

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

Whakakapia te -\frac{y}{3}+2 mō te x ki tērā atu whārite, -5x+5y=10.

$$-5\left(-\frac{1}{3}y+2\right)+5y=10$$

Whakareatia -5 ki te -\frac{y}{3}+2.

$$\frac{5}{3}y-10+5y=10$$

Tāpiri \frac{5y}{3} ki te 5y.

$$\frac{20}{3}y-10=10$$

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

$$\frac{20}{3}y=20$$

Whakawehea ngā taha e rua o te whārite ki te \frac{20}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=3$$

Whakaurua te 3 mō y ki x=-\frac{1}{3}y+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{3}\times 3+2$$

Whakareatia -\frac{1}{3} ki te 3.

$$x=-1+2$$

Tāpiri 2 ki te -1.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=3$$

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.

$$21x+7y=42,-5x+5y=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}21&7\\-5&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}42\\10\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}21&7\\-5&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}21&7\\-5&5\end{matrix}\right))\left(\begin{matrix}21&7\\-5&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}21&7\\-5&5\end{matrix}\right))\left(\begin{matrix}42\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}21&7\\-5&5\end{matrix}\right))\left(\begin{matrix}42\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}21&7\\-5&5\end{matrix}\right))\left(\begin{matrix}42\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{21\times 5-7\left(-5\right)}&-\frac{7}{21\times 5-7\left(-5\right)}\\-\frac{-5}{21\times 5-7\left(-5\right)}&\frac{21}{21\times 5-7\left(-5\right)}\end{matrix}\right)\left(\begin{matrix}42\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{28}&-\frac{1}{20}\\\frac{1}{28}&\frac{3}{20}\end{matrix}\right)\left(\begin{matrix}42\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{28}\times 42-\frac{1}{20}\times 10\\\frac{1}{28}\times 42+\frac{3}{20}\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=3$$

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.

$$21x+7y=42,-5x+5y=10$$

Kia ōrite ai a 21x me -5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 21.

$$-5\times 21x-5\times 7y=-5\times 42,21\left(-5\right)x+21\times 5y=21\times 10$$

Whakarūnātia.

$$-105x-35y=-210,-105x+105y=210$$

Me tango -105x+105y=210 mai i -105x-35y=-210 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-105x+105x-35y-105y=-210-210$$

Tāpiri -105x ki te 105x. Ka whakakore atu ngā kupu -105x me 105x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-35y-105y=-210-210$$

Tāpiri -35y ki te -105y.

$$-140y=-210-210$$

Tāpiri -210 ki te -210.

$$-140y=-420$$

Whakawehea ngā taha e rua ki te -140.

$$y=3$$

Whakaurua te 3 mō y ki -5x+5y=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-5x+5\times 3=10$$

Whakareatia 5 ki te 3.

$$-5x+15=10$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$-5x=-5$$

Whakawehea ngā taha e rua ki te -5.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=3$$