Whakaoti i te {l}{7x-8y=-12}{-4x+2y=3}

$$\left. \begin{array}{l}{ 7 x - 8 y = - 12 }\\{ - 4 x + 2 y = 3 }\end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

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.

$$7x-8y=-12,-4x+2y=3$$

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

$$7x-8y=-12$$

Me tāpiri 8y ki ngā taha e rua o te whārite.

$$7x=8y-12$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(8y-12\right)$$

Whakareatia \frac{1}{7} ki te 8y-12.

$$x=\frac{8}{7}y-\frac{12}{7}$$

Whakakapia te \frac{8y-12}{7} mō te x ki tērā atu whārite, -4x+2y=3.

$$-4\left(\frac{8}{7}y-\frac{12}{7}\right)+2y=3$$

Whakareatia -4 ki te \frac{8y-12}{7}.

$$-\frac{32}{7}y+\frac{48}{7}+2y=3$$

Tāpiri -\frac{32y}{7} ki te 2y.

$$-\frac{18}{7}y+\frac{48}{7}=3$$

Me tango \frac{48}{7} mai i ngā taha e rua o te whārite.

$$-\frac{18}{7}y=-\frac{27}{7}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{18}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{3}{2}$$

Whakaurua te \frac{3}{2} mō y ki x=\frac{8}{7}y-\frac{12}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{8}{7}\times \frac{3}{2}-\frac{12}{7}$$

Whakareatia \frac{8}{7} ki te \frac{3}{2} 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.

$$x=\frac{12-12}{7}$$

Tāpiri -\frac{12}{7} ki te \frac{12}{7} 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.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=\frac{3}{2}$$

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.

$$7x-8y=-12,-4x+2y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&-8\\-4&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-12\\3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&-8\\-4&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&-8\\-4&2\end{matrix}\right))\left(\begin{matrix}7&-8\\-4&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-8\\-4&2\end{matrix}\right))\left(\begin{matrix}-12\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-8\\-4&2\end{matrix}\right))\left(\begin{matrix}-12\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-8\\-4&2\end{matrix}\right))\left(\begin{matrix}-12\\3\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7\times 2-\left(-8\left(-4\right)\right)}&-\frac{-8}{7\times 2-\left(-8\left(-4\right)\right)}\\-\frac{-4}{7\times 2-\left(-8\left(-4\right)\right)}&\frac{7}{7\times 2-\left(-8\left(-4\right)\right)}\end{matrix}\right)\left(\begin{matrix}-12\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{9}&-\frac{4}{9}\\-\frac{2}{9}&-\frac{7}{18}\end{matrix}\right)\left(\begin{matrix}-12\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{9}\left(-12\right)-\frac{4}{9}\times 3\\-\frac{2}{9}\left(-12\right)-\frac{7}{18}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\\frac{3}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=0,y=\frac{3}{2}$$

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.

$$7x-8y=-12,-4x+2y=3$$

Kia ōrite ai a 7x 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 7.

$$-4\times 7x-4\left(-8\right)y=-4\left(-12\right),7\left(-4\right)x+7\times 2y=7\times 3$$

Whakarūnātia.

$$-28x+32y=48,-28x+14y=21$$

Me tango -28x+14y=21 mai i -28x+32y=48 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-28x+28x+32y-14y=48-21$$

Tāpiri -28x ki te 28x. Ka whakakore atu ngā kupu -28x me 28x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$32y-14y=48-21$$

Tāpiri 32y ki te -14y.

$$18y=48-21$$

Tāpiri 48 ki te -21.

$$18y=27$$

Whakawehea ngā taha e rua ki te 18.

$$y=\frac{3}{2}$$

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

$$-4x+2\times \frac{3}{2}=3$$

Whakareatia 2 ki te \frac{3}{2}.

$$-4x+3=3$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$-4x=0$$

Whakawehea ngā taha e rua ki te -4.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=\frac{3}{2}$$