Whakaoti i te {l}{3x-7y=114}{4x-3y=76}

$$\left\{ \begin{array} { l } { 3 x - 7 y = 114 } \\ { 4 x - 3 y = 76 } \end{array} \right.$$

Whakaoti mō x, y

$$x=10$$

$$y=-12$$

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.

$$3x-7y=114,4x-3y=76$$

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

$$3x-7y=114$$

Me tāpiri 7y ki ngā taha e rua o te whārite.

$$3x=7y+114$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(7y+114\right)$$

Whakareatia \frac{1}{3} ki te 7y+114.

$$x=\frac{7}{3}y+38$$

Whakakapia te \frac{7y}{3}+38 mō te x ki tērā atu whārite, 4x-3y=76.

$$4\left(\frac{7}{3}y+38\right)-3y=76$$

Whakareatia 4 ki te \frac{7y}{3}+38.

$$\frac{28}{3}y+152-3y=76$$

Tāpiri \frac{28y}{3} ki te -3y.

$$\frac{19}{3}y+152=76$$

Me tango 152 mai i ngā taha e rua o te whārite.

$$\frac{19}{3}y=-76$$

Whakawehea ngā taha e rua o te whārite ki te \frac{19}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-12$$

Whakaurua te -12 mō y ki x=\frac{7}{3}y+38. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{7}{3}\left(-12\right)+38$$

Whakareatia \frac{7}{3} ki te -12.

$$x=-28+38$$

Tāpiri 38 ki te -28.

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=-12$$

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.

$$3x-7y=114,4x-3y=76$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-7\\4&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}114\\76\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-7\\4&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-7\\4&-3\end{matrix}\right))\left(\begin{matrix}3&-7\\4&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-7\\4&-3\end{matrix}\right))\left(\begin{matrix}114\\76\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-7\\4&-3\end{matrix}\right))\left(\begin{matrix}114\\76\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-7\\4&-3\end{matrix}\right))\left(\begin{matrix}114\\76\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{3\left(-3\right)-\left(-7\times 4\right)}&-\frac{-7}{3\left(-3\right)-\left(-7\times 4\right)}\\-\frac{4}{3\left(-3\right)-\left(-7\times 4\right)}&\frac{3}{3\left(-3\right)-\left(-7\times 4\right)}\end{matrix}\right)\left(\begin{matrix}114\\76\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{19}&\frac{7}{19}\\-\frac{4}{19}&\frac{3}{19}\end{matrix}\right)\left(\begin{matrix}114\\76\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{19}\times 114+\frac{7}{19}\times 76\\-\frac{4}{19}\times 114+\frac{3}{19}\times 76\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\-12\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=10,y=-12$$

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.

$$3x-7y=114,4x-3y=76$$

Kia ōrite ai a 3x 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 3.

$$4\times 3x+4\left(-7\right)y=4\times 114,3\times 4x+3\left(-3\right)y=3\times 76$$

Whakarūnātia.

$$12x-28y=456,12x-9y=228$$

Me tango 12x-9y=228 mai i 12x-28y=456 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x-28y+9y=456-228$$

Tāpiri 12x ki te -12x. Ka whakakore atu ngā kupu 12x me -12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-28y+9y=456-228$$

Tāpiri -28y ki te 9y.

$$-19y=456-228$$

Tāpiri 456 ki te -228.

$$-19y=228$$

Whakawehea ngā taha e rua ki te -19.

$$y=-12$$

Whakaurua te -12 mō y ki 4x-3y=76. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-3\left(-12\right)=76$$

Whakareatia -3 ki te -12.

$$4x+36=76$$

Me tango 36 mai i ngā taha e rua o te whārite.

$$4x=40$$

Whakawehea ngā taha e rua ki te 4.

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=-12$$