Whakaoti i te {l}{2x-3y=0}{-x+15y=3}

$$\left. \begin{array} { l } { 2 x - 3 y = 0 } \\ { - x + 15 y = 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{1}{3}\approx 0.333333333$$

$$y=\frac{2}{9}\approx 0.222222222$$

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.

$$2x-3y=0,-x+15y=3$$

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

$$2x-3y=0$$

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

$$2x=3y$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\times 3y$$

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

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

Whakakapia te \frac{3y}{2} mō te x ki tērā atu whārite, -x+15y=3.

$$-\frac{3}{2}y+15y=3$$

Tāpiri -\frac{3y}{2} ki te 15y.

$$\frac{27}{2}y=3$$

Whakawehea ngā taha e rua o te whārite ki te \frac{27}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{2}{9}$$

Whakaurua te \frac{2}{9} 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.

$$x=\frac{3}{2}\times \frac{2}{9}$$

Whakareatia \frac{3}{2} ki te \frac{2}{9} 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.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{3},y=\frac{2}{9}$$

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.

$$2x-3y=0,-x+15y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-3\\-1&15\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-3\\-1&15\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-3\\-1&15\end{matrix}\right))\left(\begin{matrix}2&-3\\-1&15\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\-1&15\end{matrix}\right))\left(\begin{matrix}0\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\-1&15\end{matrix}\right))\left(\begin{matrix}0\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\-1&15\end{matrix}\right))\left(\begin{matrix}0\\3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{2\times 15-\left(-3\left(-1\right)\right)}&-\frac{-3}{2\times 15-\left(-3\left(-1\right)\right)}\\-\frac{-1}{2\times 15-\left(-3\left(-1\right)\right)}&\frac{2}{2\times 15-\left(-3\left(-1\right)\right)}\end{matrix}\right)\left(\begin{matrix}0\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{9}&\frac{1}{9}\\\frac{1}{27}&\frac{2}{27}\end{matrix}\right)\left(\begin{matrix}0\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}\times 3\\\frac{2}{27}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\\\frac{2}{9}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1}{3},y=\frac{2}{9}$$

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.

$$2x-3y=0,-x+15y=3$$

Kia ōrite ai a 2x 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 2.

$$-2x-\left(-3y\right)=0,2\left(-1\right)x+2\times 15y=2\times 3$$

Whakarūnātia.

$$-2x+3y=0,-2x+30y=6$$

Me tango -2x+30y=6 mai i -2x+3y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x+3y-30y=-6$$

Tāpiri -2x ki te 2x. Ka whakakore atu ngā kupu -2x me 2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3y-30y=-6$$

Tāpiri 3y ki te -30y.

$$-27y=-6$$

Whakawehea ngā taha e rua ki te -27.

$$y=\frac{2}{9}$$

Whakaurua te \frac{2}{9} mō y ki -x+15y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-x+15\times \frac{2}{9}=3$$

Whakareatia 15 ki te \frac{2}{9}.

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

Me tango \frac{10}{3} mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te -1.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{3},y=\frac{2}{9}$$