Whakaoti i te {l}{5x+2y=34}{7x-3y=7}

$$\left. \begin{array} { l } { 5 x + 2 y = 34 } \\ { 7 x - 3 y = 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=7$$

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.

$$5x+2y=34,7x-3y=7$$

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

$$5x+2y=34$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$5x=-2y+34$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-2y+34\right)$$

Whakareatia \frac{1}{5} ki te -2y+34.

$$x=-\frac{2}{5}y+\frac{34}{5}$$

Whakakapia te \frac{-2y+34}{5} mō te x ki tērā atu whārite, 7x-3y=7.

$$7\left(-\frac{2}{5}y+\frac{34}{5}\right)-3y=7$$

Whakareatia 7 ki te \frac{-2y+34}{5}.

$$-\frac{14}{5}y+\frac{238}{5}-3y=7$$

Tāpiri -\frac{14y}{5} ki te -3y.

$$-\frac{29}{5}y+\frac{238}{5}=7$$

Me tango \frac{238}{5} mai i ngā taha e rua o te whārite.

$$-\frac{29}{5}y=-\frac{203}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{29}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=7$$

Whakaurua te 7 mō y ki x=-\frac{2}{5}y+\frac{34}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{5}\times 7+\frac{34}{5}$$

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

$$x=\frac{-14+34}{5}$$

Tāpiri \frac{34}{5} ki te -\frac{14}{5} 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.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=7$$

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.

$$5x+2y=34,7x-3y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&2\\7&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}34\\7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&2\\7&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&2\\7&-3\end{matrix}\right))\left(\begin{matrix}5&2\\7&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\7&-3\end{matrix}\right))\left(\begin{matrix}34\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\7&-3\end{matrix}\right))\left(\begin{matrix}34\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\7&-3\end{matrix}\right))\left(\begin{matrix}34\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{5\left(-3\right)-2\times 7}&-\frac{2}{5\left(-3\right)-2\times 7}\\-\frac{7}{5\left(-3\right)-2\times 7}&\frac{5}{5\left(-3\right)-2\times 7}\end{matrix}\right)\left(\begin{matrix}34\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{29}&\frac{2}{29}\\\frac{7}{29}&-\frac{5}{29}\end{matrix}\right)\left(\begin{matrix}34\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{29}\times 34+\frac{2}{29}\times 7\\\frac{7}{29}\times 34-\frac{5}{29}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\7\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=7$$

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.

$$5x+2y=34,7x-3y=7$$

Kia ōrite ai a 5x me 7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$7\times 5x+7\times 2y=7\times 34,5\times 7x+5\left(-3\right)y=5\times 7$$

Whakarūnātia.

$$35x+14y=238,35x-15y=35$$

Me tango 35x-15y=35 mai i 35x+14y=238 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$35x-35x+14y+15y=238-35$$

Tāpiri 35x ki te -35x. Ka whakakore atu ngā kupu 35x me -35x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$14y+15y=238-35$$

Tāpiri 14y ki te 15y.

$$29y=238-35$$

Tāpiri 238 ki te -35.

$$29y=203$$

Whakawehea ngā taha e rua ki te 29.

$$y=7$$

Whakaurua te 7 mō y ki 7x-3y=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x-3\times 7=7$$

Whakareatia -3 ki te 7.

$$7x-21=7$$

Me tāpiri 21 ki ngā taha e rua o te whārite.

$$7x=28$$

Whakawehea ngā taha e rua ki te 7.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=7$$