Whakaoti i te {l}{7x+3y=43}{4x-3y=67}

$$\left\{ \begin{array} { l } { 7 x + 3 y = 43 } \\ { 4 x - 3 y = 67 } \end{array} \right.$$

Whakaoti mō x, y

$$x=10$$

$$y=-9$$

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.

$$7x+3y=43,4x-3y=67$$

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

$$7x+3y=43$$

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

$$7x=-3y+43$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(-3y+43\right)$$

Whakareatia \frac{1}{7} ki te -3y+43.

$$x=-\frac{3}{7}y+\frac{43}{7}$$

Whakakapia te \frac{-3y+43}{7} mō te x ki tērā atu whārite, 4x-3y=67.

$$4\left(-\frac{3}{7}y+\frac{43}{7}\right)-3y=67$$

Whakareatia 4 ki te \frac{-3y+43}{7}.

$$-\frac{12}{7}y+\frac{172}{7}-3y=67$$

Tāpiri -\frac{12y}{7} ki te -3y.

$$-\frac{33}{7}y+\frac{172}{7}=67$$

Me tango \frac{172}{7} mai i ngā taha e rua o te whārite.

$$-\frac{33}{7}y=\frac{297}{7}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{33}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-9$$

Whakaurua te -9 mō y ki x=-\frac{3}{7}y+\frac{43}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{7}\left(-9\right)+\frac{43}{7}$$

Whakareatia -\frac{3}{7} ki te -9.

$$x=\frac{27+43}{7}$$

Tāpiri \frac{43}{7} ki te \frac{27}{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.

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=-9$$

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.

$$7x+3y=43,4x-3y=67$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&3\\4&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}43\\67\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}7&3\\4&-3\end{matrix}\right))\left(\begin{matrix}7&3\\4&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&3\\4&-3\end{matrix}\right))\left(\begin{matrix}43\\67\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&3\\4&-3\end{matrix}\right))\left(\begin{matrix}43\\67\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&3\\4&-3\end{matrix}\right))\left(\begin{matrix}43\\67\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{7\left(-3\right)-3\times 4}&-\frac{3}{7\left(-3\right)-3\times 4}\\-\frac{4}{7\left(-3\right)-3\times 4}&\frac{7}{7\left(-3\right)-3\times 4}\end{matrix}\right)\left(\begin{matrix}43\\67\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}&\frac{1}{11}\\\frac{4}{33}&-\frac{7}{33}\end{matrix}\right)\left(\begin{matrix}43\\67\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}\times 43+\frac{1}{11}\times 67\\\frac{4}{33}\times 43-\frac{7}{33}\times 67\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\-9\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=10,y=-9$$

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.

$$7x+3y=43,4x-3y=67$$

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.

$$4\times 7x+4\times 3y=4\times 43,7\times 4x+7\left(-3\right)y=7\times 67$$

Whakarūnātia.

$$28x+12y=172,28x-21y=469$$

Me tango 28x-21y=469 mai i 28x+12y=172 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$28x-28x+12y+21y=172-469$$

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.

$$12y+21y=172-469$$

Tāpiri 12y ki te 21y.

$$33y=172-469$$

Tāpiri 172 ki te -469.

$$33y=-297$$

Whakawehea ngā taha e rua ki te 33.

$$y=-9$$

Whakaurua te -9 mō y ki 4x-3y=67. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-3\left(-9\right)=67$$

Whakareatia -3 ki te -9.

$$4x+27=67$$

Me tango 27 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.

$$x=10,y=-9$$