Whakaoti i te 3x+9=6ytext{and}-2x-2y-12=0

$$3 x + 9 = 6 y \text { and } - 2 x - 2 y - 12 = 0$$

Whakaoti mō x, y

$$x=-5$$

$$y=-1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 6y mai i ngā taha e rua.

$$3x+9-6y=0$$

Tangohia te 9 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$3x-6y=-9$$

Whakaarohia te whārite tuarua. Me tāpiri te 12 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$-2x-2y=12$$

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.

$$3x-6y=-9,-2x-2y=12$$

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

$$3x-6y=-9$$

Me tāpiri 6y ki ngā taha e rua o te whārite.

$$3x=6y-9$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(6y-9\right)$$

Whakareatia \frac{1}{3} ki te 6y-9.

$$x=2y-3$$

Whakakapia te 2y-3 mō te x ki tērā atu whārite, -2x-2y=12.

$$-2\left(2y-3\right)-2y=12$$

Whakareatia -2 ki te 2y-3.

$$-4y+6-2y=12$$

Tāpiri -4y ki te -2y.

$$-6y+6=12$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$-6y=6$$

Whakawehea ngā taha e rua ki te -6.

$$y=-1$$

Whakaurua te -1 mō y ki x=2y-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

Whakareatia 2 ki te -1.

$$x=-2-3$$

Tāpiri -3 ki te -2.

$$x=-5$$

Kua oti te pūnaha te whakatau.

$$x=-5,y=-1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 6y mai i ngā taha e rua.

$$3x+9-6y=0$$

Tangohia te 9 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$3x-6y=-9$$

Whakaarohia te whārite tuarua. Me tāpiri te 12 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$-2x-2y=12$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$3x-6y=-9,-2x-2y=12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-6\\-2&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-9\\12\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&-6\\-2&-2\end{matrix}\right))\left(\begin{matrix}3&-6\\-2&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-6\\-2&-2\end{matrix}\right))\left(\begin{matrix}-9\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-6\\-2&-2\end{matrix}\right))\left(\begin{matrix}-9\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-6\\-2&-2\end{matrix}\right))\left(\begin{matrix}-9\\12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{3\left(-2\right)-\left(-6\left(-2\right)\right)}&-\frac{-6}{3\left(-2\right)-\left(-6\left(-2\right)\right)}\\-\frac{-2}{3\left(-2\right)-\left(-6\left(-2\right)\right)}&\frac{3}{3\left(-2\right)-\left(-6\left(-2\right)\right)}\end{matrix}\right)\left(\begin{matrix}-9\\12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}&-\frac{1}{3}\\-\frac{1}{9}&-\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}-9\\12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}\left(-9\right)-\frac{1}{3}\times 12\\-\frac{1}{9}\left(-9\right)-\frac{1}{6}\times 12\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-5,y=-1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 6y mai i ngā taha e rua.

$$3x+9-6y=0$$

Tangohia te 9 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$3x-6y=-9$$

Whakaarohia te whārite tuarua. Me tāpiri te 12 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$-2x-2y=12$$

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.

$$3x-6y=-9,-2x-2y=12$$

Kia ōrite ai a 3x me -2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$-2\times 3x-2\left(-6\right)y=-2\left(-9\right),3\left(-2\right)x+3\left(-2\right)y=3\times 12$$

Whakarūnātia.

$$-6x+12y=18,-6x-6y=36$$

Me tango -6x-6y=36 mai i -6x+12y=18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-6x+6x+12y+6y=18-36$$

Tāpiri -6x ki te 6x. Ka whakakore atu ngā kupu -6x me 6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$12y+6y=18-36$$

Tāpiri 12y ki te 6y.

$$18y=18-36$$

Tāpiri 18 ki te -36.

$$18y=-18$$

Whakawehea ngā taha e rua ki te 18.

$$y=-1$$

Whakaurua te -1 mō y ki -2x-2y=12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x-2\left(-1\right)=12$$

Whakareatia -2 ki te -1.

$$-2x+2=12$$

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

$$-2x=10$$

Whakawehea ngā taha e rua ki te -2.

$$x=-5$$

Kua oti te pūnaha te whakatau.

$$x=-5,y=-1$$