Whakaoti i te {l}{-2x+3y=13}{6x-5y=-3}

$$\left. \begin{array} { l } { - 2 x + 3 y = 13 } \\ { 6 x - 5 y = - 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x=7$$

$$y=9$$

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.

$$-2x+3y=13,6x-5y=-3$$

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

$$-2x+3y=13$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$-2x=-3y+13$$

Whakawehea ngā taha e rua ki te -2.

$$x=-\frac{1}{2}\left(-3y+13\right)$$

Whakareatia -\frac{1}{2} ki te -3y+13.

$$x=\frac{3}{2}y-\frac{13}{2}$$

Whakakapia te \frac{3y-13}{2} mō te x ki tērā atu whārite, 6x-5y=-3.

$$6\left(\frac{3}{2}y-\frac{13}{2}\right)-5y=-3$$

Whakareatia 6 ki te \frac{3y-13}{2}.

$$9y-39-5y=-3$$

Tāpiri 9y ki te -5y.

$$4y-39=-3$$

Me tāpiri 39 ki ngā taha e rua o te whārite.

$$4y=36$$

Whakawehea ngā taha e rua ki te 4.

$$y=9$$

Whakaurua te 9 mō y ki x=\frac{3}{2}y-\frac{13}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{2}\times 9-\frac{13}{2}$$

Whakareatia \frac{3}{2} ki te 9.

$$x=\frac{27-13}{2}$$

Tāpiri -\frac{13}{2} ki te \frac{27}{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.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=9$$

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.

$$-2x+3y=13,6x-5y=-3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-2&3\\6&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}13\\-3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-2&3\\6&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}-2&3\\6&-5\end{matrix}\right))\left(\begin{matrix}-2&3\\6&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&3\\6&-5\end{matrix}\right))\left(\begin{matrix}13\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&3\\6&-5\end{matrix}\right))\left(\begin{matrix}13\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&3\\6&-5\end{matrix}\right))\left(\begin{matrix}13\\-3\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{-2\left(-5\right)-3\times 6}&-\frac{3}{-2\left(-5\right)-3\times 6}\\-\frac{6}{-2\left(-5\right)-3\times 6}&-\frac{2}{-2\left(-5\right)-3\times 6}\end{matrix}\right)\left(\begin{matrix}13\\-3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{8}&\frac{3}{8}\\\frac{3}{4}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}13\\-3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{8}\times 13+\frac{3}{8}\left(-3\right)\\\frac{3}{4}\times 13+\frac{1}{4}\left(-3\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\9\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=7,y=9$$

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.

$$-2x+3y=13,6x-5y=-3$$

Kia ōrite ai a -2x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -2.

$$6\left(-2\right)x+6\times 3y=6\times 13,-2\times 6x-2\left(-5\right)y=-2\left(-3\right)$$

Whakarūnātia.

$$-12x+18y=78,-12x+10y=6$$

Me tango -12x+10y=6 mai i -12x+18y=78 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-12x+12x+18y-10y=78-6$$

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.

$$18y-10y=78-6$$

Tāpiri 18y ki te -10y.

$$8y=78-6$$

Tāpiri 78 ki te -6.

$$8y=72$$

Whakawehea ngā taha e rua ki te 8.

$$y=9$$

Whakaurua te 9 mō y ki 6x-5y=-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x-5\times 9=-3$$

Whakareatia -5 ki te 9.

$$6x-45=-3$$

Me tāpiri 45 ki ngā taha e rua o te whārite.

$$6x=42$$

Whakawehea ngā taha e rua ki te 6.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=9$$