Whakaoti i te {l}{3x+y=0}{2x-5y=6}

$$\left. \begin{array} { l } { 3 x + y = 0 } \\ { 2 x - 5 y = 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{6}{17}\approx 0.352941176$$

$$y = -\frac{18}{17} = -1\frac{1}{17} \approx -1.058823529$$

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.

$$3x+y=0,2x-5y=6$$

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

$$3x+y=0$$

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

$$3x=-y$$

Whakawehea ngā taha e rua ki te 3.

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

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

$$x=-\frac{1}{3}y$$

Whakakapia te -\frac{y}{3} mō te x ki tērā atu whārite, 2x-5y=6.

$$2\left(-\frac{1}{3}\right)y-5y=6$$

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

$$-\frac{2}{3}y-5y=6$$

Tāpiri -\frac{2y}{3} ki te -5y.

$$-\frac{17}{3}y=6$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{17}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{18}{17}$$

Whakaurua te -\frac{18}{17} mō y ki x=-\frac{1}{3}y. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{3}\left(-\frac{18}{17}\right)$$

Whakareatia -\frac{1}{3} ki te -\frac{18}{17} 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.

$$x=\frac{6}{17}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{6}{17},y=-\frac{18}{17}$$

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.

$$3x+y=0,2x-5y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&1\\2&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&1\\2&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&1\\2&-5\end{matrix}\right))\left(\begin{matrix}3&1\\2&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\2&-5\end{matrix}\right))\left(\begin{matrix}0\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\2&-5\end{matrix}\right))\left(\begin{matrix}0\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\2&-5\end{matrix}\right))\left(\begin{matrix}0\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{3\left(-5\right)-2}&-\frac{1}{3\left(-5\right)-2}\\-\frac{2}{3\left(-5\right)-2}&\frac{3}{3\left(-5\right)-2}\end{matrix}\right)\left(\begin{matrix}0\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{17}&\frac{1}{17}\\\frac{2}{17}&-\frac{3}{17}\end{matrix}\right)\left(\begin{matrix}0\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{17}\times 6\\-\frac{3}{17}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{17}\\-\frac{18}{17}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{6}{17},y=-\frac{18}{17}$$

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.

$$3x+y=0,2x-5y=6$$

Kia ōrite ai a 3x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$2\times 3x+2y=0,3\times 2x+3\left(-5\right)y=3\times 6$$

Whakarūnātia.

$$6x+2y=0,6x-15y=18$$

Me tango 6x-15y=18 mai i 6x+2y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+2y+15y=-18$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2y+15y=-18$$

Tāpiri 2y ki te 15y.

$$17y=-18$$

Whakawehea ngā taha e rua ki te 17.

$$y=-\frac{18}{17}$$

Whakaurua te -\frac{18}{17} mō y ki 2x-5y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-5\left(-\frac{18}{17}\right)=6$$

Whakareatia -5 ki te -\frac{18}{17}.

$$2x+\frac{90}{17}=6$$

Me tango \frac{90}{17} mai i ngā taha e rua o te whārite.

$$2x=\frac{12}{17}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{6}{17}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{6}{17},y=-\frac{18}{17}$$