Whakaoti i te {l}{x=5y+5}{6x-4y=7}

$$\left\{ \begin{array} { l } { x = 5y + 5 } \\ { 6 x - 4 y = 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{15}{26}\approx 0.576923077$$

$$y=-\frac{23}{26}\approx -0.884615385$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 5y mai i ngā taha e rua.

$$x-5y=5$$

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.

$$x-5y=5,6x-4y=7$$

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

$$x-5y=5$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$x=5y+5$$

Whakakapia te 5+5y mō te x ki tērā atu whārite, 6x-4y=7.

$$6\left(5y+5\right)-4y=7$$

Whakareatia 6 ki te 5+5y.

$$30y+30-4y=7$$

Tāpiri 30y ki te -4y.

$$26y+30=7$$

Me tango 30 mai i ngā taha e rua o te whārite.

$$26y=-23$$

Whakawehea ngā taha e rua ki te 26.

$$y=-\frac{23}{26}$$

Whakaurua te -\frac{23}{26} mō y ki x=5y+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=5\left(-\frac{23}{26}\right)+5$$

Whakareatia 5 ki te -\frac{23}{26}.

$$x=-\frac{115}{26}+5$$

Tāpiri 5 ki te -\frac{115}{26}.

$$x=\frac{15}{26}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{15}{26},y=-\frac{23}{26}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 5y mai i ngā taha e rua.

$$x-5y=5$$

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

$$x-5y=5,6x-4y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-5\\6&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-5\\6&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-5\\6&-4\end{matrix}\right))\left(\begin{matrix}1&-5\\6&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-5\\6&-4\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-5\\6&-4\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-5\\6&-4\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{-4-\left(-5\times 6\right)}&-\frac{-5}{-4-\left(-5\times 6\right)}\\-\frac{6}{-4-\left(-5\times 6\right)}&\frac{1}{-4-\left(-5\times 6\right)}\end{matrix}\right)\left(\begin{matrix}5\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{13}&\frac{5}{26}\\-\frac{3}{13}&\frac{1}{26}\end{matrix}\right)\left(\begin{matrix}5\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{13}\times 5+\frac{5}{26}\times 7\\-\frac{3}{13}\times 5+\frac{1}{26}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{26}\\-\frac{23}{26}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{15}{26},y=-\frac{23}{26}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 5y mai i ngā taha e rua.

$$x-5y=5$$

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.

$$x-5y=5,6x-4y=7$$

Kia ōrite ai a x 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 1.

$$6x+6\left(-5\right)y=6\times 5,6x-4y=7$$

Whakarūnātia.

$$6x-30y=30,6x-4y=7$$

Me tango 6x-4y=7 mai i 6x-30y=30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-30y+4y=30-7$$

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.

$$-30y+4y=30-7$$

Tāpiri -30y ki te 4y.

$$-26y=30-7$$

Tāpiri 30 ki te -7.

$$-26y=23$$

Whakawehea ngā taha e rua ki te -26.

$$y=-\frac{23}{26}$$

Whakaurua te -\frac{23}{26} mō y ki 6x-4y=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x-4\left(-\frac{23}{26}\right)=7$$

Whakareatia -4 ki te -\frac{23}{26}.

$$6x+\frac{46}{13}=7$$

Me tango \frac{46}{13} mai i ngā taha e rua o te whārite.

$$6x=\frac{45}{13}$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{15}{26}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{15}{26},y=-\frac{23}{26}$$