Whakaoti i te {l}{5x-y=6}{3x-4y=-10}quad{l}{x-3}{y=-2}

$$\left. \begin{array} { l } { 5 x - y = 6 } \\ { 3 x - 4 y = - 10 } \end{array} \right. \quad \left\{ \begin{array} { l } { x - 3 } \\ { y = - 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=4$$

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.

$$5x-y=6,3x-4y=-10$$

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

$$5x-y=6$$

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

$$5x=y+6$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(y+6\right)$$

Whakareatia \frac{1}{5} ki te y+6.

$$x=\frac{1}{5}y+\frac{6}{5}$$

Whakakapia te \frac{6+y}{5} mō te x ki tērā atu whārite, 3x-4y=-10.

$$3\left(\frac{1}{5}y+\frac{6}{5}\right)-4y=-10$$

Whakareatia 3 ki te \frac{6+y}{5}.

$$\frac{3}{5}y+\frac{18}{5}-4y=-10$$

Tāpiri \frac{3y}{5} ki te -4y.

$$-\frac{17}{5}y+\frac{18}{5}=-10$$

Me tango \frac{18}{5} mai i ngā taha e rua o te whārite.

$$-\frac{17}{5}y=-\frac{68}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{17}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=4$$

Whakaurua te 4 mō y ki x=\frac{1}{5}y+\frac{6}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{5}\times 4+\frac{6}{5}$$

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

$$x=\frac{4+6}{5}$$

Tāpiri \frac{6}{5} ki te \frac{4}{5} 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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=4$$

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.

$$5x-y=6,3x-4y=-10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-1\\3&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\-10\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}5&-1\\3&-4\end{matrix}\right))\left(\begin{matrix}5&-1\\3&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\3&-4\end{matrix}\right))\left(\begin{matrix}6\\-10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\3&-4\end{matrix}\right))\left(\begin{matrix}6\\-10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\3&-4\end{matrix}\right))\left(\begin{matrix}6\\-10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{5\left(-4\right)-\left(-3\right)}&-\frac{-1}{5\left(-4\right)-\left(-3\right)}\\-\frac{3}{5\left(-4\right)-\left(-3\right)}&\frac{5}{5\left(-4\right)-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}6\\-10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{17}&-\frac{1}{17}\\\frac{3}{17}&-\frac{5}{17}\end{matrix}\right)\left(\begin{matrix}6\\-10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{17}\times 6-\frac{1}{17}\left(-10\right)\\\frac{3}{17}\times 6-\frac{5}{17}\left(-10\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=2,y=4$$

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.

$$5x-y=6,3x-4y=-10$$

Kia ōrite ai a 5x 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 5.

$$3\times 5x+3\left(-1\right)y=3\times 6,5\times 3x+5\left(-4\right)y=5\left(-10\right)$$

Whakarūnātia.

$$15x-3y=18,15x-20y=-50$$

Me tango 15x-20y=-50 mai i 15x-3y=18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x-3y+20y=18+50$$

Tāpiri 15x ki te -15x. Ka whakakore atu ngā kupu 15x me -15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-3y+20y=18+50$$

Tāpiri -3y ki te 20y.

$$17y=18+50$$

Tāpiri 18 ki te 50.

$$17y=68$$

Whakawehea ngā taha e rua ki te 17.

$$y=4$$

Whakaurua te 4 mō y ki 3x-4y=-10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-4\times 4=-10$$

Whakareatia -4 ki te 4.

$$3x-16=-10$$

Me tāpiri 16 ki ngā taha e rua o te whārite.

$$3x=6$$

Whakawehea ngā taha e rua ki te 3.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=4$$