Whakaoti i te {l}{5y=-13-4x}{3x+6y=13}

$$\left. \begin{array} { l } { 5 y = - 13 - 4 x } \\ { 3 x + 6 y = 13 } \end{array} \right.$$

Whakaoti mō y, x

$$x = -\frac{143}{9} = -15\frac{8}{9} \approx -15.888888889$$

$$y = \frac{91}{9} = 10\frac{1}{9} \approx 10.111111111$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te 4x ki ngā taha e rua.

$$5y+4x=-13$$

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.

$$5y+4x=-13,6y+3x=13$$

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

$$5y+4x=-13$$

Me tango 4x mai i ngā taha e rua o te whārite.

$$5y=-4x-13$$

Whakawehea ngā taha e rua ki te 5.

$$y=\frac{1}{5}\left(-4x-13\right)$$

Whakareatia \frac{1}{5} ki te -4x-13.

$$y=-\frac{4}{5}x-\frac{13}{5}$$

Whakakapia te \frac{-4x-13}{5} mō te y ki tērā atu whārite, 6y+3x=13.

$$6\left(-\frac{4}{5}x-\frac{13}{5}\right)+3x=13$$

Whakareatia 6 ki te \frac{-4x-13}{5}.

$$-\frac{24}{5}x-\frac{78}{5}+3x=13$$

Tāpiri -\frac{24x}{5} ki te 3x.

$$-\frac{9}{5}x-\frac{78}{5}=13$$

Me tāpiri \frac{78}{5} ki ngā taha e rua o te whārite.

$$-\frac{9}{5}x=\frac{143}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{9}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-\frac{143}{9}$$

Whakaurua te -\frac{143}{9} mō x ki y=-\frac{4}{5}x-\frac{13}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-\frac{4}{5}\left(-\frac{143}{9}\right)-\frac{13}{5}$$

Whakareatia -\frac{4}{5} ki te -\frac{143}{9} 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.

$$y=\frac{572}{45}-\frac{13}{5}$$

Tāpiri -\frac{13}{5} ki te \frac{572}{45} 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.

$$y=\frac{91}{9}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{91}{9},x=-\frac{143}{9}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te 4x ki ngā taha e rua.

$$5y+4x=-13$$

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

$$5y+4x=-13,6y+3x=13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&4\\6&3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-13\\13\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&4\\6&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&4\\6&3\end{matrix}\right))\left(\begin{matrix}5&4\\6&3\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}5&4\\6&3\end{matrix}\right))\left(\begin{matrix}-13\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}5&4\\6&3\end{matrix}\right))\left(\begin{matrix}-13\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}5&4\\6&3\end{matrix}\right))\left(\begin{matrix}-13\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5\times 3-4\times 6}&-\frac{4}{5\times 3-4\times 6}\\-\frac{6}{5\times 3-4\times 6}&\frac{5}{5\times 3-4\times 6}\end{matrix}\right)\left(\begin{matrix}-13\\13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&\frac{4}{9}\\\frac{2}{3}&-\frac{5}{9}\end{matrix}\right)\left(\begin{matrix}-13\\13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}\left(-13\right)+\frac{4}{9}\times 13\\\frac{2}{3}\left(-13\right)-\frac{5}{9}\times 13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{91}{9}\\-\frac{143}{9}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=\frac{91}{9},x=-\frac{143}{9}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te 4x ki ngā taha e rua.

$$5y+4x=-13$$

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.

$$5y+4x=-13,6y+3x=13$$

Kia ōrite ai a 5y me 6y, 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 5.

$$6\times 5y+6\times 4x=6\left(-13\right),5\times 6y+5\times 3x=5\times 13$$

Whakarūnātia.

$$30y+24x=-78,30y+15x=65$$

Me tango 30y+15x=65 mai i 30y+24x=-78 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$30y-30y+24x-15x=-78-65$$

Tāpiri 30y ki te -30y. Ka whakakore atu ngā kupu 30y me -30y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$24x-15x=-78-65$$

Tāpiri 24x ki te -15x.

$$9x=-78-65$$

Tāpiri -78 ki te -65.

$$9x=-143$$

Whakawehea ngā taha e rua ki te 9.

$$x=-\frac{143}{9}$$

Whakaurua te -\frac{143}{9} mō x ki 6y+3x=13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$6y+3\left(-\frac{143}{9}\right)=13$$

Whakareatia 3 ki te -\frac{143}{9}.

$$6y-\frac{143}{3}=13$$

Me tāpiri \frac{143}{3} ki ngā taha e rua o te whārite.

$$6y=\frac{182}{3}$$

Whakawehea ngā taha e rua ki te 6.

$$y=\frac{91}{9}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{91}{9},x=-\frac{143}{9}$$