Whakaoti i te {l}{10x+2y=-78}{-3x-2y=-29}

$$\left.\begin{array} { l } { 10 x + 2 y = - 78 } \\ { - 3 x - 2 y = - 29 } \end{array} \right\}$$

Whakaoti mō x, y

$$x = -\frac{107}{7} = -15\frac{2}{7} \approx -15.285714286$$

$$y = \frac{262}{7} = 37\frac{3}{7} \approx 37.428571429$$

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.

$$10x+2y=-78,-3x-2y=-29$$

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

$$10x+2y=-78$$

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

$$10x=-2y-78$$

Whakawehea ngā taha e rua ki te 10.

$$x=\frac{1}{10}\left(-2y-78\right)$$

Whakareatia \frac{1}{10} ki te -2y-78.

$$x=-\frac{1}{5}y-\frac{39}{5}$$

Whakakapia te \frac{-y-39}{5} mō te x ki tērā atu whārite, -3x-2y=-29.

$$-3\left(-\frac{1}{5}y-\frac{39}{5}\right)-2y=-29$$

Whakareatia -3 ki te \frac{-y-39}{5}.

$$\frac{3}{5}y+\frac{117}{5}-2y=-29$$

Tāpiri \frac{3y}{5} ki te -2y.

$$-\frac{7}{5}y+\frac{117}{5}=-29$$

Me tango \frac{117}{5} mai i ngā taha e rua o te whārite.

$$-\frac{7}{5}y=-\frac{262}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{7}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{262}{7}$$

Whakaurua te \frac{262}{7} mō y ki x=-\frac{1}{5}y-\frac{39}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{5}\times \frac{262}{7}-\frac{39}{5}$$

Whakareatia -\frac{1}{5} ki te \frac{262}{7} 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.

$$x=-\frac{262}{35}-\frac{39}{5}$$

Tāpiri -\frac{39}{5} ki te -\frac{262}{35} 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.

$$x=-\frac{107}{7}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{107}{7},y=\frac{262}{7}$$

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.

$$10x+2y=-78,-3x-2y=-29$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}10&2\\-3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-78\\-29\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}10&2\\-3&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}10&2\\-3&-2\end{matrix}\right))\left(\begin{matrix}10&2\\-3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&2\\-3&-2\end{matrix}\right))\left(\begin{matrix}-78\\-29\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&2\\-3&-2\end{matrix}\right))\left(\begin{matrix}-78\\-29\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&2\\-3&-2\end{matrix}\right))\left(\begin{matrix}-78\\-29\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{10\left(-2\right)-2\left(-3\right)}&-\frac{2}{10\left(-2\right)-2\left(-3\right)}\\-\frac{-3}{10\left(-2\right)-2\left(-3\right)}&\frac{10}{10\left(-2\right)-2\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}-78\\-29\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}&\frac{1}{7}\\-\frac{3}{14}&-\frac{5}{7}\end{matrix}\right)\left(\begin{matrix}-78\\-29\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}\left(-78\right)+\frac{1}{7}\left(-29\right)\\-\frac{3}{14}\left(-78\right)-\frac{5}{7}\left(-29\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{107}{7}\\\frac{262}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{107}{7},y=\frac{262}{7}$$

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.

$$10x+2y=-78,-3x-2y=-29$$

Kia ōrite ai a 10x 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 10.

$$-3\times 10x-3\times 2y=-3\left(-78\right),10\left(-3\right)x+10\left(-2\right)y=10\left(-29\right)$$

Whakarūnātia.

$$-30x-6y=234,-30x-20y=-290$$

Me tango -30x-20y=-290 mai i -30x-6y=234 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-30x+30x-6y+20y=234+290$$

Tāpiri -30x ki te 30x. Ka whakakore atu ngā kupu -30x me 30x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-6y+20y=234+290$$

Tāpiri -6y ki te 20y.

$$14y=234+290$$

Tāpiri 234 ki te 290.

$$14y=524$$

Whakawehea ngā taha e rua ki te 14.

$$y=\frac{262}{7}$$

Whakaurua te \frac{262}{7} mō y ki -3x-2y=-29. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-3x-2\times \frac{262}{7}=-29$$

Whakareatia -2 ki te \frac{262}{7}.

$$-3x-\frac{524}{7}=-29$$

Me tāpiri \frac{524}{7} ki ngā taha e rua o te whārite.

$$-3x=\frac{321}{7}$$

Whakawehea ngā taha e rua ki te -3.

$$x=-\frac{107}{7}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{107}{7},y=\frac{262}{7}$$