Whakaoti i te {r}{-7x+2y=-39}{9x-5y=55}

$$\left. \begin{array} { r } { - 7 x + 2 y = - 39 } \\ { 9 x - 5 y = 55 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=-2$$

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.

$$-7x+2y=-39,9x-5y=55$$

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

$$-7x+2y=-39$$

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

$$-7x=-2y-39$$

Whakawehea ngā taha e rua ki te -7.

$$x=-\frac{1}{7}\left(-2y-39\right)$$

Whakareatia -\frac{1}{7} ki te -2y-39.

$$x=\frac{2}{7}y+\frac{39}{7}$$

Whakakapia te \frac{2y+39}{7} mō te x ki tērā atu whārite, 9x-5y=55.

$$9\left(\frac{2}{7}y+\frac{39}{7}\right)-5y=55$$

Whakareatia 9 ki te \frac{2y+39}{7}.

$$\frac{18}{7}y+\frac{351}{7}-5y=55$$

Tāpiri \frac{18y}{7} ki te -5y.

$$-\frac{17}{7}y+\frac{351}{7}=55$$

Me tango \frac{351}{7} mai i ngā taha e rua o te whārite.

$$-\frac{17}{7}y=\frac{34}{7}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{17}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-2$$

Whakaurua te -2 mō y ki x=\frac{2}{7}y+\frac{39}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{7}\left(-2\right)+\frac{39}{7}$$

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

$$x=\frac{-4+39}{7}$$

Tāpiri \frac{39}{7} ki te -\frac{4}{7} 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.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=-2$$

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.

$$-7x+2y=-39,9x-5y=55$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-7&2\\9&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-39\\55\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-7&2\\9&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}-7&2\\9&-5\end{matrix}\right))\left(\begin{matrix}-7&2\\9&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&2\\9&-5\end{matrix}\right))\left(\begin{matrix}-39\\55\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&2\\9&-5\end{matrix}\right))\left(\begin{matrix}-39\\55\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&2\\9&-5\end{matrix}\right))\left(\begin{matrix}-39\\55\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{-7\left(-5\right)-2\times 9}&-\frac{2}{-7\left(-5\right)-2\times 9}\\-\frac{9}{-7\left(-5\right)-2\times 9}&-\frac{7}{-7\left(-5\right)-2\times 9}\end{matrix}\right)\left(\begin{matrix}-39\\55\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{17}&-\frac{2}{17}\\-\frac{9}{17}&-\frac{7}{17}\end{matrix}\right)\left(\begin{matrix}-39\\55\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{17}\left(-39\right)-\frac{2}{17}\times 55\\-\frac{9}{17}\left(-39\right)-\frac{7}{17}\times 55\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=5,y=-2$$

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.

$$-7x+2y=-39,9x-5y=55$$

Kia ōrite ai a -7x me 9x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 9 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -7.

$$9\left(-7\right)x+9\times 2y=9\left(-39\right),-7\times 9x-7\left(-5\right)y=-7\times 55$$

Whakarūnātia.

$$-63x+18y=-351,-63x+35y=-385$$

Me tango -63x+35y=-385 mai i -63x+18y=-351 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-63x+63x+18y-35y=-351+385$$

Tāpiri -63x ki te 63x. Ka whakakore atu ngā kupu -63x me 63x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$18y-35y=-351+385$$

Tāpiri 18y ki te -35y.

$$-17y=-351+385$$

Tāpiri -351 ki te 385.

$$-17y=34$$

Whakawehea ngā taha e rua ki te -17.

$$y=-2$$

Whakaurua te -2 mō y ki 9x-5y=55. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$9x-5\left(-2\right)=55$$

Whakareatia -5 ki te -2.

$$9x+10=55$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$9x=45$$

Whakawehea ngā taha e rua ki te 9.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=-2$$