Whakaoti i te {l}{5x+y=-17}{2x+5y=7}

$$\left. \begin{array} { l } { 5 x + y = - 17 } \\ { 2 x + 5 y = 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-4$$

$$y=3$$

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.

$$5x+y=-17,2x+5y=7$$

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

$$5x+y=-17$$

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

$$5x=-y-17$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-y-17\right)$$

Whakareatia \frac{1}{5} ki te -y-17.

$$x=-\frac{1}{5}y-\frac{17}{5}$$

Whakakapia te \frac{-y-17}{5} mō te x ki tērā atu whārite, 2x+5y=7.

$$2\left(-\frac{1}{5}y-\frac{17}{5}\right)+5y=7$$

Whakareatia 2 ki te \frac{-y-17}{5}.

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

Tāpiri -\frac{2y}{5} ki te 5y.

$$\frac{23}{5}y-\frac{34}{5}=7$$

Me tāpiri \frac{34}{5} ki ngā taha e rua o te whārite.

$$\frac{23}{5}y=\frac{69}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{23}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=3$$

Whakaurua te 3 mō y ki x=-\frac{1}{5}y-\frac{17}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{5}\times 3-\frac{17}{5}$$

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

$$x=\frac{-3-17}{5}$$

Tāpiri -\frac{17}{5} ki te -\frac{3}{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.

$$x=-4,y=3$$

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.

$$5x+y=-17,2x+5y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&1\\2&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-17\\7\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}5&1\\2&5\end{matrix}\right))\left(\begin{matrix}5&1\\2&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\2&5\end{matrix}\right))\left(\begin{matrix}-17\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\2&5\end{matrix}\right))\left(\begin{matrix}-17\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\2&5\end{matrix}\right))\left(\begin{matrix}-17\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5\times 5-2}&-\frac{1}{5\times 5-2}\\-\frac{2}{5\times 5-2}&\frac{5}{5\times 5-2}\end{matrix}\right)\left(\begin{matrix}-17\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{23}&-\frac{1}{23}\\-\frac{2}{23}&\frac{5}{23}\end{matrix}\right)\left(\begin{matrix}-17\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{23}\left(-17\right)-\frac{1}{23}\times 7\\-\frac{2}{23}\left(-17\right)+\frac{5}{23}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-4,y=3$$

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.

$$5x+y=-17,2x+5y=7$$

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.

$$2\times 5x+2y=2\left(-17\right),5\times 2x+5\times 5y=5\times 7$$

Whakarūnātia.

$$10x+2y=-34,10x+25y=35$$

Me tango 10x+25y=35 mai i 10x+2y=-34 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x+2y-25y=-34-35$$

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.

$$2y-25y=-34-35$$

Tāpiri 2y ki te -25y.

$$-23y=-34-35$$

Tāpiri -34 ki te -35.

$$-23y=-69$$

Whakawehea ngā taha e rua ki te -23.

$$y=3$$

Whakaurua te 3 mō y ki 2x+5y=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+5\times 3=7$$

Whakareatia 5 ki te 3.

$$2x+15=7$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$2x=-8$$

Whakawehea ngā taha e rua ki te 2.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=3$$