Whakaoti i te {l}{2x-3y=-1}{2x+3y=16}

$$\left. \begin{array} { l } { 2 x - 3 y = - 1 } \\ { 2 x + 3 y = 16 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{15}{4} = 3\frac{3}{4} = 3.75$$

$$y = \frac{17}{6} = 2\frac{5}{6} \approx 2.833333333$$

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.

$$2x-3y=-1,2x+3y=16$$

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

$$2x-3y=-1$$

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

$$2x=3y-1$$

Whakawehea ngā taha e rua ki te 2.

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

Whakareatia \frac{1}{2} ki te 3y-1.

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

Whakakapia te \frac{3y-1}{2} mō te x ki tērā atu whārite, 2x+3y=16.

$$2\left(\frac{3}{2}y-\frac{1}{2}\right)+3y=16$$

Whakareatia 2 ki te \frac{3y-1}{2}.

$$3y-1+3y=16$$

Tāpiri 3y ki te 3y.

$$6y-1=16$$

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

$$6y=17$$

Whakawehea ngā taha e rua ki te 6.

$$y=\frac{17}{6}$$

Whakaurua te \frac{17}{6} mō y ki x=\frac{3}{2}y-\frac{1}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{2}\times \frac{17}{6}-\frac{1}{2}$$

Whakareatia \frac{3}{2} ki te \frac{17}{6} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{17}{4}-\frac{1}{2}$$

Tāpiri -\frac{1}{2} ki te \frac{17}{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.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{15}{4},y=\frac{17}{6}$$

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.

$$2x-3y=-1,2x+3y=16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-3\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\16\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&-3\\2&3\end{matrix}\right))\left(\begin{matrix}2&-3\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\2&3\end{matrix}\right))\left(\begin{matrix}-1\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\2&3\end{matrix}\right))\left(\begin{matrix}-1\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\2&3\end{matrix}\right))\left(\begin{matrix}-1\\16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2\times 3-\left(-3\times 2\right)}&-\frac{-3}{2\times 3-\left(-3\times 2\right)}\\-\frac{2}{2\times 3-\left(-3\times 2\right)}&\frac{2}{2\times 3-\left(-3\times 2\right)}\end{matrix}\right)\left(\begin{matrix}-1\\16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&\frac{1}{4}\\-\frac{1}{6}&\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}-1\\16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\left(-1\right)+\frac{1}{4}\times 16\\-\frac{1}{6}\left(-1\right)+\frac{1}{6}\times 16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{4}\\\frac{17}{6}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{15}{4},y=\frac{17}{6}$$

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.

$$2x-3y=-1,2x+3y=16$$

Me tango 2x+3y=16 mai i 2x-3y=-1 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x-3y-3y=-1-16$$

Tāpiri 2x ki te -2x. Ka whakakore atu ngā kupu 2x me -2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-3y-3y=-1-16$$

Tāpiri -3y ki te -3y.

$$-6y=-1-16$$

Tāpiri -1 ki te -16.

$$-6y=-17$$

Whakawehea ngā taha e rua ki te -6.

$$y=\frac{17}{6}$$

Whakaurua te \frac{17}{6} mō y ki 2x+3y=16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+3\times \frac{17}{6}=16$$

Whakareatia 3 ki te \frac{17}{6}.

$$2x+\frac{17}{2}=16$$

Me tango \frac{17}{2} mai i ngā taha e rua o te whārite.

$$2x=\frac{15}{2}$$

Whakawehea ngā taha e rua ki te 2.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{15}{4},y=\frac{17}{6}$$