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

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

Whakaoti mō x, y

$$x=-\frac{a}{12}-\frac{5}{4}$$

$$y=\frac{-a-3}{4}$$

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.

$$3x-5y=a,6x-2y=-6$$

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

$$3x-5y=a$$

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

$$3x=5y+a$$

Whakawehea ngā taha e rua ki te 3.

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

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

$$x=\frac{5}{3}y+\frac{a}{3}$$

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

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

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

$$10y+2a-2y=-6$$

Tāpiri 10y ki te -2y.

$$8y+2a=-6$$

Me tango 2a mai i ngā taha e rua o te whārite.

$$8y=-2a-6$$

Whakawehea ngā taha e rua ki te 8.

$$y=\frac{-a-3}{4}$$

Whakaurua te \frac{-3-a}{4} mō y ki x=\frac{5}{3}y+\frac{a}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{3}\times \frac{-a-3}{4}+\frac{a}{3}$$

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

$$x=-\frac{5a}{12}-\frac{5}{4}+\frac{a}{3}$$

Tāpiri \frac{a}{3} ki te -\frac{5}{4}-\frac{5a}{12}.

$$x=-\frac{a}{12}-\frac{5}{4}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{a}{12}-\frac{5}{4},y=\frac{-a-3}{4}$$

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.

$$3x-5y=a,6x-2y=-6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{12}&\frac{5}{24}\\-\frac{1}{4}&\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}a\\-6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{12}a+\frac{5}{24}\left(-6\right)\\-\frac{1}{4}a+\frac{1}{8}\left(-6\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{a}{12}-\frac{5}{4}\\\frac{-a-3}{4}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{a}{12}-\frac{5}{4},y=\frac{-a-3}{4}$$

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.

$$3x-5y=a,6x-2y=-6$$

Kia ōrite ai a 3x 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 3.

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

Whakarūnātia.

$$18x-30y=6a,18x-6y=-18$$

Me tango 18x-6y=-18 mai i 18x-30y=6a mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$18x-18x-30y+6y=6a+18$$

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.

$$-30y+6y=6a+18$$

Tāpiri -30y ki te 6y.

$$-24y=6a+18$$

Whakawehea ngā taha e rua ki te -24.

$$y=\frac{-a-3}{4}$$

Whakaurua te \frac{-3-a}{4} mō y ki 6x-2y=-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x-2\times \frac{-a-3}{4}=-6$$

Whakareatia -2 ki te \frac{-3-a}{4}.

$$6x+\frac{a+3}{2}=-6$$

Me tango \frac{3+a}{2} mai i ngā taha e rua o te whārite.

$$6x=\frac{-a-15}{2}$$

Whakawehea ngā taha e rua ki te 6.

$$x=-\frac{a}{12}-\frac{5}{4}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{a}{12}-\frac{5}{4},y=\frac{-a-3}{4}$$