Whakaoti i te {l}{-5x-3y-9=0}{4x-18y-54=0}

$$\left. \begin{array} { l } { - 5 x - 3 y - 9 = 0 } \\ { 4 x - 18 y - 54 = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=-3$$

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.

$$-5x-3y-9=0,4x-18y-54=0$$

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

$$-5x-3y-9=0$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$-5x-3y=9$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$-5x=3y+9$$

Whakawehea ngā taha e rua ki te -5.

$$x=-\frac{1}{5}\left(3y+9\right)$$

Whakareatia -\frac{1}{5} ki te 9+3y.

$$x=-\frac{3}{5}y-\frac{9}{5}$$

Whakakapia te \frac{-3y-9}{5} mō te x ki tērā atu whārite, 4x-18y-54=0.

$$4\left(-\frac{3}{5}y-\frac{9}{5}\right)-18y-54=0$$

Whakareatia 4 ki te \frac{-3y-9}{5}.

$$-\frac{12}{5}y-\frac{36}{5}-18y-54=0$$

Tāpiri -\frac{12y}{5} ki te -18y.

$$-\frac{102}{5}y-\frac{36}{5}-54=0$$

Tāpiri -\frac{36}{5} ki te -54.

$$-\frac{102}{5}y-\frac{306}{5}=0$$

Me tāpiri \frac{306}{5} ki ngā taha e rua o te whārite.

$$-\frac{102}{5}y=\frac{306}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{102}{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{3}{5}y-\frac{9}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{5}\left(-3\right)-\frac{9}{5}$$

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

$$x=\frac{9-9}{5}$$

Tāpiri -\frac{9}{5} ki te \frac{9}{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.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-3$$

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.

$$-5x-3y-9=0,4x-18y-54=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-5&-3\\4&-18\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\54\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-5&-3\\4&-18\end{matrix}\right).

$$inverse(\left(\begin{matrix}-5&-3\\4&-18\end{matrix}\right))\left(\begin{matrix}-5&-3\\4&-18\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&-3\\4&-18\end{matrix}\right))\left(\begin{matrix}9\\54\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&-3\\4&-18\end{matrix}\right))\left(\begin{matrix}9\\54\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&-3\\4&-18\end{matrix}\right))\left(\begin{matrix}9\\54\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{18}{-5\left(-18\right)-\left(-3\times 4\right)}&-\frac{-3}{-5\left(-18\right)-\left(-3\times 4\right)}\\-\frac{4}{-5\left(-18\right)-\left(-3\times 4\right)}&-\frac{5}{-5\left(-18\right)-\left(-3\times 4\right)}\end{matrix}\right)\left(\begin{matrix}9\\54\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{17}&\frac{1}{34}\\-\frac{2}{51}&-\frac{5}{102}\end{matrix}\right)\left(\begin{matrix}9\\54\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{17}\times 9+\frac{1}{34}\times 54\\-\frac{2}{51}\times 9-\frac{5}{102}\times 54\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\-3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=0,y=-3$$

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.

$$-5x-3y-9=0,4x-18y-54=0$$

Kia ōrite ai a -5x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -5.

$$4\left(-5\right)x+4\left(-3\right)y+4\left(-9\right)=0,-5\times 4x-5\left(-18\right)y-5\left(-54\right)=0$$

Whakarūnātia.

$$-20x-12y-36=0,-20x+90y+270=0$$

Me tango -20x+90y+270=0 mai i -20x-12y-36=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-20x+20x-12y-90y-36-270=0$$

Tāpiri -20x ki te 20x. Ka whakakore atu ngā kupu -20x me 20x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-12y-90y-36-270=0$$

Tāpiri -12y ki te -90y.

$$-102y-36-270=0$$

Tāpiri -36 ki te -270.

$$-102y-306=0$$

Me tāpiri 306 ki ngā taha e rua o te whārite.

$$-102y=306$$

Whakawehea ngā taha e rua ki te -102.

$$y=-3$$

Whakaurua te -3 mō y ki 4x-18y-54=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-18\left(-3\right)-54=0$$

Whakareatia -18 ki te -3.

$$4x+54-54=0$$

Tāpiri 54 ki te -54.

$$4x=0$$

Whakawehea ngā taha e rua ki te 4.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-3$$