Whakaoti i te {c}{-3x+5y=2}{x+10y=-24}

$$\left. \begin{array} { c } { - 3 x + 5 y = 2 } \\ { x + 10 y = - 24 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-4$$

$$y=-2$$

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.

$$-3x+5y=2,x+10y=-24$$

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

$$-3x+5y=2$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$-3x=-5y+2$$

Whakawehea ngā taha e rua ki te -3.

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

Whakareatia -\frac{1}{3} ki te -5y+2.

$$x=\frac{5}{3}y-\frac{2}{3}$$

Whakakapia te \frac{5y-2}{3} mō te x ki tērā atu whārite, x+10y=-24.

$$\frac{5}{3}y-\frac{2}{3}+10y=-24$$

Tāpiri \frac{5y}{3} ki te 10y.

$$\frac{35}{3}y-\frac{2}{3}=-24$$

Me tāpiri \frac{2}{3} ki ngā taha e rua o te whārite.

$$\frac{35}{3}y=-\frac{70}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{35}{3}, 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{5}{3}y-\frac{2}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{3}\left(-2\right)-\frac{2}{3}$$

Whakareatia \frac{5}{3} ki te -2.

$$x=\frac{-10-2}{3}$$

Tāpiri -\frac{2}{3} ki te -\frac{10}{3} 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.

$$x=-4,y=-2$$

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.

$$-3x+5y=2,x+10y=-24$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-3&5\\1&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\-24\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-3&5\\1&10\end{matrix}\right).

$$inverse(\left(\begin{matrix}-3&5\\1&10\end{matrix}\right))\left(\begin{matrix}-3&5\\1&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&5\\1&10\end{matrix}\right))\left(\begin{matrix}2\\-24\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&5\\1&10\end{matrix}\right))\left(\begin{matrix}2\\-24\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&5\\1&10\end{matrix}\right))\left(\begin{matrix}2\\-24\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{-3\times 10-5}&-\frac{5}{-3\times 10-5}\\-\frac{1}{-3\times 10-5}&-\frac{3}{-3\times 10-5}\end{matrix}\right)\left(\begin{matrix}2\\-24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{7}&\frac{1}{7}\\\frac{1}{35}&\frac{3}{35}\end{matrix}\right)\left(\begin{matrix}2\\-24\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{7}\times 2+\frac{1}{7}\left(-24\right)\\\frac{1}{35}\times 2+\frac{3}{35}\left(-24\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\\-2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-4,y=-2$$

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.

$$-3x+5y=2,x+10y=-24$$

Kia ōrite ai a -3x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -3.

$$-3x+5y=2,-3x-3\times 10y=-3\left(-24\right)$$

Whakarūnātia.

$$-3x+5y=2,-3x-30y=72$$

Me tango -3x-30y=72 mai i -3x+5y=2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-3x+3x+5y+30y=2-72$$

Tāpiri -3x ki te 3x. Ka whakakore atu ngā kupu -3x me 3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$5y+30y=2-72$$

Tāpiri 5y ki te 30y.

$$35y=2-72$$

Tāpiri 2 ki te -72.

$$35y=-70$$

Whakawehea ngā taha e rua ki te 35.

$$y=-2$$

Whakaurua te -2 mō y ki x+10y=-24. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+10\left(-2\right)=-24$$

Whakareatia 10 ki te -2.

$$x-20=-24$$

Me tāpiri 20 ki ngā taha e rua o te whārite.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=-2$$