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

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

Whakaoti mō x, y

$$x=-1$$

$$y=1$$

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.

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

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

$$x-5y=-6$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$x=5y-6$$

Whakakapia te 5y-6 mō te x ki tērā atu whārite, -4x+y=5.

$$-4\left(5y-6\right)+y=5$$

Whakareatia -4 ki te 5y-6.

$$-20y+24+y=5$$

Tāpiri -20y ki te y.

$$-19y+24=5$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$-19y=-19$$

Whakawehea ngā taha e rua ki te -19.

$$y=1$$

Whakaurua te 1 mō y ki x=5y-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=5-6$$

Tāpiri -6 ki te 5.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=1$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{19}&-\frac{5}{19}\\-\frac{4}{19}&-\frac{1}{19}\end{matrix}\right)\left(\begin{matrix}-6\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{19}\left(-6\right)-\frac{5}{19}\times 5\\-\frac{4}{19}\left(-6\right)-\frac{1}{19}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=1$$

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.

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

Kia ōrite ai a x me -4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$-4x-4\left(-5\right)y=-4\left(-6\right),-4x+y=5$$

Whakarūnātia.

$$-4x+20y=24,-4x+y=5$$

Me tango -4x+y=5 mai i -4x+20y=24 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-4x+4x+20y-y=24-5$$

Tāpiri -4x ki te 4x. Ka whakakore atu ngā kupu -4x me 4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$20y-y=24-5$$

Tāpiri 20y ki te -y.

$$19y=24-5$$

Tāpiri 24 ki te -5.

$$19y=19$$

Whakawehea ngā taha e rua ki te 19.

$$y=1$$

Whakaurua te 1 mō y ki -4x+y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-4x+1=5$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$-4x=4$$

Whakawehea ngā taha e rua ki te -4.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=1$$