Whakaoti i te {l}{-4x+y=-15}{2x-3y=5}

$$\left\{ \begin{array} { l } { - 4 x + y = - 15 } \\ { 2 x - 3 y = 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=1$$

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.

$$-4x+y=-15,2x-3y=5$$

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

$$-4x+y=-15$$

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

$$-4x=-y-15$$

Whakawehea ngā taha e rua ki te -4.

$$x=-\frac{1}{4}\left(-y-15\right)$$

Whakareatia -\frac{1}{4} ki te -y-15.

$$x=\frac{1}{4}y+\frac{15}{4}$$

Whakakapia te \frac{15+y}{4} mō te x ki tērā atu whārite, 2x-3y=5.

$$2\left(\frac{1}{4}y+\frac{15}{4}\right)-3y=5$$

Whakareatia 2 ki te \frac{15+y}{4}.

$$\frac{1}{2}y+\frac{15}{2}-3y=5$$

Tāpiri \frac{y}{2} ki te -3y.

$$-\frac{5}{2}y+\frac{15}{2}=5$$

Me tango \frac{15}{2} mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua o te whārite ki te -\frac{5}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

Whakaurua te 1 mō y ki x=\frac{1}{4}y+\frac{15}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1+15}{4}$$

Tāpiri \frac{15}{4} ki te \frac{1}{4} 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.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=1$$

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.

$$-4x+y=-15,2x-3y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-4&1\\2&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-15\\5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-4&1\\2&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}-4&1\\2&-3\end{matrix}\right))\left(\begin{matrix}-4&1\\2&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&1\\2&-3\end{matrix}\right))\left(\begin{matrix}-15\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&1\\2&-3\end{matrix}\right))\left(\begin{matrix}-15\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&1\\2&-3\end{matrix}\right))\left(\begin{matrix}-15\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{-4\left(-3\right)-2}&-\frac{1}{-4\left(-3\right)-2}\\-\frac{2}{-4\left(-3\right)-2}&-\frac{4}{-4\left(-3\right)-2}\end{matrix}\right)\left(\begin{matrix}-15\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{10}&-\frac{1}{10}\\-\frac{1}{5}&-\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}-15\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{10}\left(-15\right)-\frac{1}{10}\times 5\\-\frac{1}{5}\left(-15\right)-\frac{2}{5}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=1$$

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.

$$-4x+y=-15,2x-3y=5$$

Kia ōrite ai a -4x 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 -4.

$$2\left(-4\right)x+2y=2\left(-15\right),-4\times 2x-4\left(-3\right)y=-4\times 5$$

Whakarūnātia.

$$-8x+2y=-30,-8x+12y=-20$$

Me tango -8x+12y=-20 mai i -8x+2y=-30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-8x+8x+2y-12y=-30+20$$

Tāpiri -8x ki te 8x. Ka whakakore atu ngā kupu -8x me 8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2y-12y=-30+20$$

Tāpiri 2y ki te -12y.

$$-10y=-30+20$$

Tāpiri -30 ki te 20.

$$-10y=-10$$

Whakawehea ngā taha e rua ki te -10.

$$y=1$$

Whakaurua te 1 mō y ki 2x-3y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-3=5$$

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

$$2x=8$$

Whakawehea ngā taha e rua ki te 2.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=1$$