Whakaoti i te {l}{5x+2y=0}{6x-1y=2}

$$\left. \begin{array} { l } { 5 x + 2 y = 0 } \\ { 6 x - 1 y = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{4}{17}\approx 0.235294118$$

$$y=-\frac{10}{17}\approx -0.588235294$$

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.

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

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

$$5x+2y=0$$

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

$$5x=-2y$$

Whakawehea ngā taha e rua ki te 5.

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

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

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

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

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

Whakareatia 6 ki te -\frac{2y}{5}.

$$-\frac{12}{5}y-y=2$$

Tāpiri -\frac{12y}{5} ki te -y.

$$-\frac{17}{5}y=2$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{17}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{10}{17}$$

Whakaurua te -\frac{10}{17} mō y ki x=-\frac{2}{5}y. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{5}\left(-\frac{10}{17}\right)$$

Whakareatia -\frac{2}{5} ki te -\frac{10}{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.

$$x=\frac{4}{17}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{4}{17},y=-\frac{10}{17}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&2\\6&-1\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{17}\times 2\\-\frac{5}{17}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{17}\\-\frac{10}{17}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{4}{17},y=-\frac{10}{17}$$

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.

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

Kia ōrite ai a 5x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$6\times 5x+6\times 2y=0,5\times 6x+5\left(-1\right)y=5\times 2$$

Whakarūnātia.

$$30x+12y=0,30x-5y=10$$

Me tango 30x-5y=10 mai i 30x+12y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$30x-30x+12y+5y=-10$$

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.

$$12y+5y=-10$$

Tāpiri 12y ki te 5y.

$$17y=-10$$

Whakawehea ngā taha e rua ki te 17.

$$y=-\frac{10}{17}$$

Whakaurua te -\frac{10}{17} mō y ki 6x-y=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x-\left(-\frac{10}{17}\right)=2$$

Me tango \frac{10}{17} mai i ngā taha e rua o te whārite.

$$6x=\frac{24}{17}$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{4}{17}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{4}{17},y=-\frac{10}{17}$$