Whakaoti i te {l}{5x+3y=7}{y=2x+1}

$$\left. \begin{array} { l } { 5 x + 3 y = 7 } \\ { y = 2 x + 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{4}{11}\approx 0.363636364$$

$$y = \frac{19}{11} = 1\frac{8}{11} \approx 1.727272727$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=1$$

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.

$$5x+3y=7,-2x+y=1$$

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

$$5x+3y=7$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$5x=-3y+7$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-3y+7\right)$$

Whakareatia \frac{1}{5} ki te -3y+7.

$$x=-\frac{3}{5}y+\frac{7}{5}$$

Whakakapia te \frac{-3y+7}{5} mō te x ki tērā atu whārite, -2x+y=1.

$$-2\left(-\frac{3}{5}y+\frac{7}{5}\right)+y=1$$

Whakareatia -2 ki te \frac{-3y+7}{5}.

$$\frac{6}{5}y-\frac{14}{5}+y=1$$

Tāpiri \frac{6y}{5} ki te y.

$$\frac{11}{5}y-\frac{14}{5}=1$$

Me tāpiri \frac{14}{5} ki ngā taha e rua o te whārite.

$$\frac{11}{5}y=\frac{19}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{11}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{19}{11}$$

Whakaurua te \frac{19}{11} mō y ki x=-\frac{3}{5}y+\frac{7}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{5}\times \frac{19}{11}+\frac{7}{5}$$

Whakareatia -\frac{3}{5} ki te \frac{19}{11} 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.

$$x=-\frac{57}{55}+\frac{7}{5}$$

Tāpiri \frac{7}{5} ki te -\frac{57}{55} 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.

$$x=\frac{4}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{4}{11},y=\frac{19}{11}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=1$$

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

$$5x+3y=7,-2x+y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&3\\-2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&3\\-2&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&3\\-2&1\end{matrix}\right))\left(\begin{matrix}5&3\\-2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&3\\-2&1\end{matrix}\right))\left(\begin{matrix}7\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&3\\-2&1\end{matrix}\right))\left(\begin{matrix}7\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&3\\-2&1\end{matrix}\right))\left(\begin{matrix}7\\1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5-3\left(-2\right)}&-\frac{3}{5-3\left(-2\right)}\\-\frac{-2}{5-3\left(-2\right)}&\frac{5}{5-3\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}7\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}&-\frac{3}{11}\\\frac{2}{11}&\frac{5}{11}\end{matrix}\right)\left(\begin{matrix}7\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}\times 7-\frac{3}{11}\\\frac{2}{11}\times 7+\frac{5}{11}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{11}\\\frac{19}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{4}{11},y=\frac{19}{11}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=1$$

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.

$$5x+3y=7,-2x+y=1$$

Kia ōrite ai a 5x 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 5.

$$-2\times 5x-2\times 3y=-2\times 7,5\left(-2\right)x+5y=5$$

Whakarūnātia.

$$-10x-6y=-14,-10x+5y=5$$

Me tango -10x+5y=5 mai i -10x-6y=-14 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-10x+10x-6y-5y=-14-5$$

Tāpiri -10x ki te 10x. Ka whakakore atu ngā kupu -10x me 10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-6y-5y=-14-5$$

Tāpiri -6y ki te -5y.

$$-11y=-14-5$$

Tāpiri -14 ki te -5.

$$-11y=-19$$

Whakawehea ngā taha e rua ki te -11.

$$y=\frac{19}{11}$$

Whakaurua te \frac{19}{11} mō y ki -2x+y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x+\frac{19}{11}=1$$

Me tango \frac{19}{11} mai i ngā taha e rua o te whārite.

$$-2x=-\frac{8}{11}$$

Whakawehea ngā taha e rua ki te -2.

$$x=\frac{4}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{4}{11},y=\frac{19}{11}$$