Whakaoti i te {l}{-4x+y=6}{-5x-y=21}

$$\left. \begin{array} { l } { - 4 x + y = 6 } \\ { - 5 x - y = 21 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-3$$

$$y=-6$$

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.

$$-4x+y=6,-5x-y=21$$

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

$$-4x+y=6$$

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

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

Whakawehea ngā taha e rua ki te -4.

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

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

$$x=\frac{1}{4}y-\frac{3}{2}$$

Whakakapia te \frac{y}{4}-\frac{3}{2} mō te x ki tērā atu whārite, -5x-y=21.

$$-5\left(\frac{1}{4}y-\frac{3}{2}\right)-y=21$$

Whakareatia -5 ki te \frac{y}{4}-\frac{3}{2}.

$$-\frac{5}{4}y+\frac{15}{2}-y=21$$

Tāpiri -\frac{5y}{4} ki te -y.

$$-\frac{9}{4}y+\frac{15}{2}=21$$

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

$$-\frac{9}{4}y=\frac{27}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{9}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-6$$

Whakaurua te -6 mō y ki x=\frac{1}{4}y-\frac{3}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{4}\left(-6\right)-\frac{3}{2}$$

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

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

Tāpiri -\frac{3}{2} ki te -\frac{3}{2} 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.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=-6$$

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.

$$-4x+y=6,-5x-y=21$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-4&1\\-5&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\21\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}-4&1\\-5&-1\end{matrix}\right))\left(\begin{matrix}-4&1\\-5&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&1\\-5&-1\end{matrix}\right))\left(\begin{matrix}6\\21\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&1\\-5&-1\end{matrix}\right))\left(\begin{matrix}6\\21\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&1\\-5&-1\end{matrix}\right))\left(\begin{matrix}6\\21\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-4\left(-1\right)-\left(-5\right)}&-\frac{1}{-4\left(-1\right)-\left(-5\right)}\\-\frac{-5}{-4\left(-1\right)-\left(-5\right)}&-\frac{4}{-4\left(-1\right)-\left(-5\right)}\end{matrix}\right)\left(\begin{matrix}6\\21\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{9}&-\frac{1}{9}\\\frac{5}{9}&-\frac{4}{9}\end{matrix}\right)\left(\begin{matrix}6\\21\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{9}\times 6-\frac{1}{9}\times 21\\\frac{5}{9}\times 6-\frac{4}{9}\times 21\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-3,y=-6$$

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.

$$-4x+y=6,-5x-y=21$$

Kia ōrite ai a -4x me -5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -4.

$$-5\left(-4\right)x-5y=-5\times 6,-4\left(-5\right)x-4\left(-1\right)y=-4\times 21$$

Whakarūnātia.

$$20x-5y=-30,20x+4y=-84$$

Me tango 20x+4y=-84 mai i 20x-5y=-30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$20x-20x-5y-4y=-30+84$$

Tāpiri 20x ki te -20x. Ka whakakore atu ngā kupu 20x me -20x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-5y-4y=-30+84$$

Tāpiri -5y ki te -4y.

$$-9y=-30+84$$

Tāpiri -30 ki te 84.

$$-9y=54$$

Whakawehea ngā taha e rua ki te -9.

$$y=-6$$

Whakaurua te -6 mō y ki -5x-y=21. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-5x-\left(-6\right)=21$$

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

$$-5x=15$$

Whakawehea ngā taha e rua ki te -5.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=-6$$