Whakaoti i te {l}{5x-4y=-3}{3x-4y=-13}

$$\left\{ \begin{array} { l } { 5 x - 4 y = - 3 } \\ { 3 x - 4 y = - 13 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=7$$

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.

$$5x-4y=-3,3x-4y=-13$$

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

$$5x-4y=-3$$

Me tāpiri 4y ki ngā taha e rua o te whārite.

$$5x=4y-3$$

Whakawehea ngā taha e rua ki te 5.

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

Whakareatia \frac{1}{5} ki te 4y-3.

$$x=\frac{4}{5}y-\frac{3}{5}$$

Whakakapia te \frac{4y-3}{5} mō te x ki tērā atu whārite, 3x-4y=-13.

$$3\left(\frac{4}{5}y-\frac{3}{5}\right)-4y=-13$$

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

$$\frac{12}{5}y-\frac{9}{5}-4y=-13$$

Tāpiri \frac{12y}{5} ki te -4y.

$$-\frac{8}{5}y-\frac{9}{5}=-13$$

Me tāpiri \frac{9}{5} ki ngā taha e rua o te whārite.

$$-\frac{8}{5}y=-\frac{56}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{8}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=7$$

Whakaurua te 7 mō y ki x=\frac{4}{5}y-\frac{3}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{4}{5}\times 7-\frac{3}{5}$$

Whakareatia \frac{4}{5} ki te 7.

$$x=\frac{28-3}{5}$$

Tāpiri -\frac{3}{5} ki te \frac{28}{5} 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.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=7$$

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.

$$5x-4y=-3,3x-4y=-13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-4\\3&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\-13\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-4\\3&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-4\\3&-4\end{matrix}\right))\left(\begin{matrix}5&-4\\3&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\3&-4\end{matrix}\right))\left(\begin{matrix}-3\\-13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\3&-4\end{matrix}\right))\left(\begin{matrix}-3\\-13\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\3&-4\end{matrix}\right))\left(\begin{matrix}-3\\-13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{5\left(-4\right)-\left(-4\times 3\right)}&-\frac{-4}{5\left(-4\right)-\left(-4\times 3\right)}\\-\frac{3}{5\left(-4\right)-\left(-4\times 3\right)}&\frac{5}{5\left(-4\right)-\left(-4\times 3\right)}\end{matrix}\right)\left(\begin{matrix}-3\\-13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&-\frac{1}{2}\\\frac{3}{8}&-\frac{5}{8}\end{matrix}\right)\left(\begin{matrix}-3\\-13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\left(-3\right)-\frac{1}{2}\left(-13\right)\\\frac{3}{8}\left(-3\right)-\frac{5}{8}\left(-13\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\7\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=7$$

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.

$$5x-4y=-3,3x-4y=-13$$

Me tango 3x-4y=-13 mai i 5x-4y=-3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5x-3x-4y+4y=-3+13$$

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

$$5x-3x=-3+13$$

Tāpiri 5x ki te -3x.

$$2x=-3+13$$

Tāpiri -3 ki te 13.

$$2x=10$$

Whakawehea ngā taha e rua ki te 2.

$$x=5$$

Whakaurua te 5 mō x ki 3x-4y=-13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$3\times 5-4y=-13$$

Whakareatia 3 ki te 5.

$$15-4y=-13$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$-4y=-28$$

Whakawehea ngā taha e rua ki te -4.

$$y=7$$

Kua oti te pūnaha te whakatau.

$$x=5,y=7$$