Whakaoti i te {r}{6x+5y=23}{4x+y=13}

$$\left. \begin{array} { r } { 6 x + 5 y = 23 } \\ { 4 x + y = 13 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=1$$

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.

$$6x+5y=23,4x+y=13$$

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

$$6x+5y=23$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$6x=-5y+23$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(-5y+23\right)$$

Whakareatia \frac{1}{6} ki te -5y+23.

$$x=-\frac{5}{6}y+\frac{23}{6}$$

Whakakapia te \frac{-5y+23}{6} mō te x ki tērā atu whārite, 4x+y=13.

$$4\left(-\frac{5}{6}y+\frac{23}{6}\right)+y=13$$

Whakareatia 4 ki te \frac{-5y+23}{6}.

$$-\frac{10}{3}y+\frac{46}{3}+y=13$$

Tāpiri -\frac{10y}{3} ki te y.

$$-\frac{7}{3}y+\frac{46}{3}=13$$

Me tango \frac{46}{3} mai i ngā taha e rua o te whārite.

$$-\frac{7}{3}y=-\frac{7}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{7}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

Whakaurua te 1 mō y ki x=-\frac{5}{6}y+\frac{23}{6}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{-5+23}{6}$$

Tāpiri \frac{23}{6} ki te -\frac{5}{6} 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.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$

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.

$$6x+5y=23,4x+y=13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&5\\4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}23\\13\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&5\\4&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&5\\4&1\end{matrix}\right))\left(\begin{matrix}6&5\\4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&5\\4&1\end{matrix}\right))\left(\begin{matrix}23\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&5\\4&1\end{matrix}\right))\left(\begin{matrix}23\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&5\\4&1\end{matrix}\right))\left(\begin{matrix}23\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6-5\times 4}&-\frac{5}{6-5\times 4}\\-\frac{4}{6-5\times 4}&\frac{6}{6-5\times 4}\end{matrix}\right)\left(\begin{matrix}23\\13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{14}&\frac{5}{14}\\\frac{2}{7}&-\frac{3}{7}\end{matrix}\right)\left(\begin{matrix}23\\13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{14}\times 23+\frac{5}{14}\times 13\\\frac{2}{7}\times 23-\frac{3}{7}\times 13\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=1$$

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.

$$6x+5y=23,4x+y=13$$

Kia ōrite ai a 6x 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 6.

$$4\times 6x+4\times 5y=4\times 23,6\times 4x+6y=6\times 13$$

Whakarūnātia.

$$24x+20y=92,24x+6y=78$$

Me tango 24x+6y=78 mai i 24x+20y=92 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$24x-24x+20y-6y=92-78$$

Tāpiri 24x ki te -24x. Ka whakakore atu ngā kupu 24x me -24x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$20y-6y=92-78$$

Tāpiri 20y ki te -6y.

$$14y=92-78$$

Tāpiri 92 ki te -78.

$$14y=14$$

Whakawehea ngā taha e rua ki te 14.

$$y=1$$

Whakaurua te 1 mō y ki 4x+y=13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+1=13$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$4x=12$$

Whakawehea ngā taha e rua ki te 4.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$