Whakaoti i te {l}{3x+5y=-1}{3x+y=3}

$$\left\{ \begin{array} { l } { 3 x + 5 y = - 1 } \\ { 3 x + y = 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

$$y=-1$$

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.

$$3x+5y=-1,3x+y=3$$

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

$$3x+5y=-1$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$3x=-5y-1$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-5y-1\right)$$

Whakareatia \frac{1}{3} ki te -5y-1.

$$x=-\frac{5}{3}y-\frac{1}{3}$$

Whakakapia te \frac{-5y-1}{3} mō te x ki tērā atu whārite, 3x+y=3.

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

Whakareatia 3 ki te \frac{-5y-1}{3}.

$$-5y-1+y=3$$

Tāpiri -5y ki te y.

$$-4y-1=3$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$-4y=4$$

Whakawehea ngā taha e rua ki te -4.

$$y=-1$$

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

$$x=-\frac{5}{3}\left(-1\right)-\frac{1}{3}$$

Whakareatia -\frac{5}{3} ki te -1.

$$x=\frac{5-1}{3}$$

Tāpiri -\frac{1}{3} ki te \frac{5}{3} 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.

$$x=\frac{4}{3}$$

Kua oti te pūnaha te whakatau.

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

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.

$$3x+5y=-1,3x+y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&5\\3&1\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{12}&\frac{5}{12}\\\frac{1}{4}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}-1\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{12}\left(-1\right)+\frac{5}{12}\times 3\\\frac{1}{4}\left(-1\right)-\frac{1}{4}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$3x+5y=-1,3x+y=3$$

Me tango 3x+y=3 mai i 3x+5y=-1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+5y-y=-1-3$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$5y-y=-1-3$$

Tāpiri 5y ki te -y.

$$4y=-1-3$$

Tāpiri -1 ki te -3.

$$4y=-4$$

Whakawehea ngā taha e rua ki te 4.

$$y=-1$$

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

$$3x-1=3$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$3x=4$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{4}{3}$$

Kua oti te pūnaha te whakatau.

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