Whakaoti i te {r}{6x-5y=14}{-3x+5y=-2}

$$\left\{ \begin{array} { r } { 6 x - 5 y = 14 } \\ { - 3 x + 5 y = - 2 } \end{array} \right.$$

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.

$$6x-5y=14,-3x+5y=-2$$

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

$$6x-5y=14$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$6x=5y+14$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(5y+14\right)$$

Whakareatia \frac{1}{6} ki te 5y+14.

$$x=\frac{5}{6}y+\frac{7}{3}$$

Whakakapia te \frac{5y}{6}+\frac{7}{3} mō te x ki tērā atu whārite, -3x+5y=-2.

$$-3\left(\frac{5}{6}y+\frac{7}{3}\right)+5y=-2$$

Whakareatia -3 ki te \frac{5y}{6}+\frac{7}{3}.

$$-\frac{5}{2}y-7+5y=-2$$

Tāpiri -\frac{5y}{2} ki te 5y.

$$\frac{5}{2}y-7=-2$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$\frac{5}{2}y=5$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{2}, 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}{6}y+\frac{7}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{6}\times 2+\frac{7}{3}$$

Whakareatia \frac{5}{6} ki te 2.

$$x=\frac{5+7}{3}$$

Tāpiri \frac{7}{3} ki te \frac{5}{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.

$$x=4$$

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.

$$6x-5y=14,-3x+5y=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&-5\\-3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14\\-2\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}6&-5\\-3&5\end{matrix}\right))\left(\begin{matrix}6&-5\\-3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-5\\-3&5\end{matrix}\right))\left(\begin{matrix}14\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-5\\-3&5\end{matrix}\right))\left(\begin{matrix}14\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-5\\-3&5\end{matrix}\right))\left(\begin{matrix}14\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{6\times 5-\left(-5\left(-3\right)\right)}&-\frac{-5}{6\times 5-\left(-5\left(-3\right)\right)}\\-\frac{-3}{6\times 5-\left(-5\left(-3\right)\right)}&\frac{6}{6\times 5-\left(-5\left(-3\right)\right)}\end{matrix}\right)\left(\begin{matrix}14\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&\frac{1}{3}\\\frac{1}{5}&\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}14\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 14+\frac{1}{3}\left(-2\right)\\\frac{1}{5}\times 14+\frac{2}{5}\left(-2\right)\end{matrix}\right)$$

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.

$$6x-5y=14,-3x+5y=-2$$

Kia ōrite ai a 6x 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 6.

$$-3\times 6x-3\left(-5\right)y=-3\times 14,6\left(-3\right)x+6\times 5y=6\left(-2\right)$$

Whakarūnātia.

$$-18x+15y=-42,-18x+30y=-12$$

Me tango -18x+30y=-12 mai i -18x+15y=-42 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-18x+18x+15y-30y=-42+12$$

Tāpiri -18x ki te 18x. Ka whakakore atu ngā kupu -18x me 18x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$15y-30y=-42+12$$

Tāpiri 15y ki te -30y.

$$-15y=-42+12$$

Tāpiri -42 ki te 12.

$$-15y=-30$$

Whakawehea ngā taha e rua ki te -15.

$$y=2$$

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.

$$-3x+5\times 2=-2$$

Whakareatia 5 ki te 2.

$$-3x+10=-2$$

Me tango 10 mai i 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.

$$x=4,y=2$$