Whakaoti i te {l}{3x+2y=-1}{6x+6y=-5}

$$\left. \begin{array} { l } { 3 x + 2 y = - 1 } \\ { 6 x + 6 y = - 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{2}{3}\approx 0.666666667$$

$$y = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

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.

$$3x+2y=-1,6x+6y=-5$$

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

$$3x+2y=-1$$

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

$$3x=-2y-1$$

Whakawehea ngā taha e rua ki te 3.

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

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

$$x=-\frac{2}{3}y-\frac{1}{3}$$

Whakakapia te \frac{-2y-1}{3} mō te x ki tērā atu whārite, 6x+6y=-5.

$$6\left(-\frac{2}{3}y-\frac{1}{3}\right)+6y=-5$$

Whakareatia 6 ki te \frac{-2y-1}{3}.

$$-4y-2+6y=-5$$

Tāpiri -4y ki te 6y.

$$2y-2=-5$$

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

$$2y=-3$$

Whakawehea ngā taha e rua ki te 2.

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

Whakaurua te -\frac{3}{2} mō y ki x=-\frac{2}{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.

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

Whakareatia -\frac{2}{3} ki te -\frac{3}{2} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

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

Tāpiri -\frac{1}{3} ki te 1.

$$x=\frac{2}{3}$$

Kua oti te pūnaha te whakatau.

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

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.

$$3x+2y=-1,6x+6y=-5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$3x+2y=-1,6x+6y=-5$$

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.

$$6\times 3x+6\times 2y=6\left(-1\right),3\times 6x+3\times 6y=3\left(-5\right)$$

Whakarūnātia.

$$18x+12y=-6,18x+18y=-15$$

Me tango 18x+18y=-15 mai i 18x+12y=-6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$18x-18x+12y-18y=-6+15$$

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.

$$12y-18y=-6+15$$

Tāpiri 12y ki te -18y.

$$-6y=-6+15$$

Tāpiri -6 ki te 15.

$$-6y=9$$

Whakawehea ngā taha e rua ki te -6.

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

Whakaurua te -\frac{3}{2} mō y ki 6x+6y=-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

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

$$6x-9=-5$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$6x=4$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{2}{3}$$

Kua oti te pūnaha te whakatau.

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