Whakaoti i te {l}{3x-13=-y}{2x+9y=-8}

$$\left. \begin{array} { l } { 3 x - 13 = - y } \\ { 2 x + 9 y = - 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=-2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te y ki ngā taha e rua.

$$3x-13+y=0$$

Me tāpiri te 13 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3x+y=13$$

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.

$$3x+y=13,2x+9y=-8$$

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

$$3x+y=13$$

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

$$3x=-y+13$$

Whakawehea ngā taha e rua ki te 3.

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

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

$$x=-\frac{1}{3}y+\frac{13}{3}$$

Whakakapia te \frac{-y+13}{3} mō te x ki tērā atu whārite, 2x+9y=-8.

$$2\left(-\frac{1}{3}y+\frac{13}{3}\right)+9y=-8$$

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

$$-\frac{2}{3}y+\frac{26}{3}+9y=-8$$

Tāpiri -\frac{2y}{3} ki te 9y.

$$\frac{25}{3}y+\frac{26}{3}=-8$$

Me tango \frac{26}{3} mai i ngā taha e rua o te whārite.

$$\frac{25}{3}y=-\frac{50}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{25}{3}, 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{1}{3}y+\frac{13}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

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

$$x=\frac{2+13}{3}$$

Tāpiri \frac{13}{3} ki te \frac{2}{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.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=-2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te y ki ngā taha e rua.

$$3x-13+y=0$$

Me tāpiri te 13 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3x+y=13$$

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

$$3x+y=13,2x+9y=-8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&1\\2&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}13\\-8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&1\\2&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&1\\2&9\end{matrix}\right))\left(\begin{matrix}3&1\\2&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\2&9\end{matrix}\right))\left(\begin{matrix}13\\-8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\2&9\end{matrix}\right))\left(\begin{matrix}13\\-8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\2&9\end{matrix}\right))\left(\begin{matrix}13\\-8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{3\times 9-2}&-\frac{1}{3\times 9-2}\\-\frac{2}{3\times 9-2}&\frac{3}{3\times 9-2}\end{matrix}\right)\left(\begin{matrix}13\\-8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{25}&-\frac{1}{25}\\-\frac{2}{25}&\frac{3}{25}\end{matrix}\right)\left(\begin{matrix}13\\-8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{25}\times 13-\frac{1}{25}\left(-8\right)\\-\frac{2}{25}\times 13+\frac{3}{25}\left(-8\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=5,y=-2$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te y ki ngā taha e rua.

$$3x-13+y=0$$

Me tāpiri te 13 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3x+y=13$$

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.

$$3x+y=13,2x+9y=-8$$

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.

$$2\times 3x+2y=2\times 13,3\times 2x+3\times 9y=3\left(-8\right)$$

Whakarūnātia.

$$6x+2y=26,6x+27y=-24$$

Me tango 6x+27y=-24 mai i 6x+2y=26 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+2y-27y=26+24$$

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.

$$2y-27y=26+24$$

Tāpiri 2y ki te -27y.

$$-25y=26+24$$

Tāpiri 26 ki te 24.

$$-25y=50$$

Whakawehea ngā taha e rua ki te -25.

$$y=-2$$

Whakaurua te -2 mō y ki 2x+9y=-8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+9\left(-2\right)=-8$$

Whakareatia 9 ki te -2.

$$2x-18=-8$$

Me tāpiri 18 ki 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.

$$x=5,y=-2$$