Whakaoti i te {l}{4x+6y=0}{x-5y=-2}

$$\left. \begin{array} { l } { 4 x + 6 y = 0 } \\ { x - 5 y = - 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{6}{13}\approx -0.461538462$$

$$y=\frac{4}{13}\approx 0.307692308$$

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.

$$4x+6y=0,x-5y=-2$$

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

$$4x+6y=0$$

Me tango 6y mai i ngā taha e rua o te whārite.

$$4x=-6y$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-6\right)y$$

Whakareatia \frac{1}{4} ki te -6y.

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

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

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

Tāpiri -\frac{3y}{2} ki te -5y.

$$-\frac{13}{2}y=-2$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{13}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{4}{13}$$

Whakaurua te \frac{4}{13} mō y ki x=-\frac{3}{2}y. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times \frac{4}{13}$$

Whakareatia -\frac{3}{2} ki te \frac{4}{13} 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.

$$x=-\frac{6}{13}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{6}{13},y=\frac{4}{13}$$

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.

$$4x+6y=0,x-5y=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&6\\1&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\-2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&6\\1&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&6\\1&-5\end{matrix}\right))\left(\begin{matrix}4&6\\1&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&6\\1&-5\end{matrix}\right))\left(\begin{matrix}0\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&6\\1&-5\end{matrix}\right))\left(\begin{matrix}0\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&6\\1&-5\end{matrix}\right))\left(\begin{matrix}0\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{4\left(-5\right)-6}&-\frac{6}{4\left(-5\right)-6}\\-\frac{1}{4\left(-5\right)-6}&\frac{4}{4\left(-5\right)-6}\end{matrix}\right)\left(\begin{matrix}0\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{26}&\frac{3}{13}\\\frac{1}{26}&-\frac{2}{13}\end{matrix}\right)\left(\begin{matrix}0\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{13}\left(-2\right)\\-\frac{2}{13}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{13}\\\frac{4}{13}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{6}{13},y=\frac{4}{13}$$

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.

$$4x+6y=0,x-5y=-2$$

Kia ōrite ai a 4x 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 4.

$$4x+6y=0,4x+4\left(-5\right)y=4\left(-2\right)$$

Whakarūnātia.

$$4x+6y=0,4x-20y=-8$$

Me tango 4x-20y=-8 mai i 4x+6y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-4x+6y+20y=8$$

Tāpiri 4x ki te -4x. Ka whakakore atu ngā kupu 4x me -4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y+20y=8$$

Tāpiri 6y ki te 20y.

$$26y=8$$

Whakawehea ngā taha e rua ki te 26.

$$y=\frac{4}{13}$$

Whakaurua te \frac{4}{13} mō y ki x-5y=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-5\times \frac{4}{13}=-2$$

Whakareatia -5 ki te \frac{4}{13}.

$$x-\frac{20}{13}=-2$$

Me tāpiri \frac{20}{13} ki ngā taha e rua o te whārite.

$$x=-\frac{6}{13}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{6}{13},y=\frac{4}{13}$$