Whakaoti i te {l}{-3x+5y=1}{4x-y=10}

$$\left\{ \begin{array} { l } { - 3 x + 5 y = 1 } \\ { 4 x - y = 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=2$$

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.

$$-3x+5y=1,4x-y=10$$

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

$$-3x+5y=1$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$-3x=-5y+1$$

Whakawehea ngā taha e rua ki te -3.

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

Whakareatia -\frac{1}{3} ki te -5y+1.

$$x=\frac{5}{3}y-\frac{1}{3}$$

Whakakapia te \frac{5y-1}{3} mō te x ki tērā atu whārite, 4x-y=10.

$$4\left(\frac{5}{3}y-\frac{1}{3}\right)-y=10$$

Whakareatia 4 ki te \frac{5y-1}{3}.

$$\frac{20}{3}y-\frac{4}{3}-y=10$$

Tāpiri \frac{20y}{3} ki te -y.

$$\frac{17}{3}y-\frac{4}{3}=10$$

Me tāpiri \frac{4}{3} ki ngā taha e rua o te whārite.

$$\frac{17}{3}y=\frac{34}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{17}{3}, 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{5}{3}y-\frac{1}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{3}\times 2-\frac{1}{3}$$

Whakareatia \frac{5}{3} ki te 2.

$$x=\frac{10-1}{3}$$

Tāpiri -\frac{1}{3} ki te \frac{10}{3} 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.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=2$$

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.

$$-3x+5y=1,4x-y=10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-3&5\\4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\10\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}-3&5\\4&-1\end{matrix}\right))\left(\begin{matrix}-3&5\\4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&5\\4&-1\end{matrix}\right))\left(\begin{matrix}1\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&5\\4&-1\end{matrix}\right))\left(\begin{matrix}1\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&5\\4&-1\end{matrix}\right))\left(\begin{matrix}1\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-3\left(-1\right)-5\times 4}&-\frac{5}{-3\left(-1\right)-5\times 4}\\-\frac{4}{-3\left(-1\right)-5\times 4}&-\frac{3}{-3\left(-1\right)-5\times 4}\end{matrix}\right)\left(\begin{matrix}1\\10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{17}&\frac{5}{17}\\\frac{4}{17}&\frac{3}{17}\end{matrix}\right)\left(\begin{matrix}1\\10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{17}+\frac{5}{17}\times 10\\\frac{4}{17}+\frac{3}{17}\times 10\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=2$$

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.

$$-3x+5y=1,4x-y=10$$

Kia ōrite ai a -3x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -3.

$$4\left(-3\right)x+4\times 5y=4,-3\times 4x-3\left(-1\right)y=-3\times 10$$

Whakarūnātia.

$$-12x+20y=4,-12x+3y=-30$$

Me tango -12x+3y=-30 mai i -12x+20y=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-12x+12x+20y-3y=4+30$$

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.

$$20y-3y=4+30$$

Tāpiri 20y ki te -3y.

$$17y=4+30$$

Tāpiri 4 ki te 30.

$$17y=34$$

Whakawehea ngā taha e rua ki te 17.

$$y=2$$

Whakaurua te 2 mō y ki 4x-y=10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-2=10$$

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

$$4x=12$$

Whakawehea ngā taha e rua ki te 4.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=2$$