Whakaoti i te {l}{7x-10y=36}{-3x+29y=14}

$$\left. \begin{array} { l } { 7 x - 10 y = 36 } \\ { - 3 x + 29 y = 14 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{1184}{173} = 6\frac{146}{173} \approx 6.843930636$$

$$y = \frac{206}{173} = 1\frac{33}{173} \approx 1.190751445$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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.

$$7x-10y=36,-3x+29y=14$$

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

$$7x-10y=36$$

Me tāpiri 10y ki ngā taha e rua o te whārite.

$$7x=10y+36$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(10y+36\right)$$

Whakareatia \frac{1}{7} ki te 10y+36.

$$x=\frac{10}{7}y+\frac{36}{7}$$

Whakakapia te \frac{10y+36}{7} mō te x ki tērā atu whārite, -3x+29y=14.

$$-3\left(\frac{10}{7}y+\frac{36}{7}\right)+29y=14$$

Whakareatia -3 ki te \frac{10y+36}{7}.

$$-\frac{30}{7}y-\frac{108}{7}+29y=14$$

Tāpiri -\frac{30y}{7} ki te 29y.

$$\frac{173}{7}y-\frac{108}{7}=14$$

Me tāpiri \frac{108}{7} ki ngā taha e rua o te whārite.

$$\frac{173}{7}y=\frac{206}{7}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{173}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{206}{173}$$

Whakaurua te \frac{206}{173} mō y ki x=\frac{10}{7}y+\frac{36}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{10}{7}\times \frac{206}{173}+\frac{36}{7}$$

Whakareatia \frac{10}{7} ki te \frac{206}{173} 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.

$$x=\frac{2060}{1211}+\frac{36}{7}$$

Tāpiri \frac{36}{7} ki te \frac{2060}{1211} 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.

$$x=\frac{1184}{173}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1184}{173},y=\frac{206}{173}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$7x-10y=36,-3x+29y=14$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&-10\\-3&29\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}36\\14\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&-10\\-3&29\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&-10\\-3&29\end{matrix}\right))\left(\begin{matrix}7&-10\\-3&29\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-10\\-3&29\end{matrix}\right))\left(\begin{matrix}36\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-10\\-3&29\end{matrix}\right))\left(\begin{matrix}36\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-10\\-3&29\end{matrix}\right))\left(\begin{matrix}36\\14\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{29}{7\times 29-\left(-10\left(-3\right)\right)}&-\frac{-10}{7\times 29-\left(-10\left(-3\right)\right)}\\-\frac{-3}{7\times 29-\left(-10\left(-3\right)\right)}&\frac{7}{7\times 29-\left(-10\left(-3\right)\right)}\end{matrix}\right)\left(\begin{matrix}36\\14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{29}{173}&\frac{10}{173}\\\frac{3}{173}&\frac{7}{173}\end{matrix}\right)\left(\begin{matrix}36\\14\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{29}{173}\times 36+\frac{10}{173}\times 14\\\frac{3}{173}\times 36+\frac{7}{173}\times 14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1184}{173}\\\frac{206}{173}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1184}{173},y=\frac{206}{173}$$

Ngā Upane e Whakamahi i te Tangohanga

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.

$$7x-10y=36,-3x+29y=14$$

Kia ōrite ai a 7x me -3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 7.

$$-3\times 7x-3\left(-10\right)y=-3\times 36,7\left(-3\right)x+7\times 29y=7\times 14$$

Whakarūnātia.

$$-21x+30y=-108,-21x+203y=98$$

Me tango -21x+203y=98 mai i -21x+30y=-108 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-21x+21x+30y-203y=-108-98$$

Tāpiri -21x ki te 21x. Ka whakakore atu ngā kupu -21x me 21x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$30y-203y=-108-98$$

Tāpiri 30y ki te -203y.

$$-173y=-108-98$$

Tāpiri -108 ki te -98.

$$-173y=-206$$

Whakawehea ngā taha e rua ki te -173.

$$y=\frac{206}{173}$$

Whakaurua te \frac{206}{173} mō y ki -3x+29y=14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-3x+29\times \frac{206}{173}=14$$

Whakareatia 29 ki te \frac{206}{173}.

$$-3x+\frac{5974}{173}=14$$

Me tango \frac{5974}{173} mai i ngā taha e rua o te whārite.

$$-3x=-\frac{3552}{173}$$

Whakawehea ngā taha e rua ki te -3.

$$x=\frac{1184}{173}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1184}{173},y=\frac{206}{173}$$