Whakaoti i te {l}{5x-3y-4=34}{5y-3x-18=34}

$$\left. \begin{array} { l } { 5 x - 3 y - 4 = 34 } \\ { 5 y - 3 x - 18 = 34 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{173}{8} = 21\frac{5}{8} = 21.625$$

$$y = \frac{187}{8} = 23\frac{3}{8} = 23.375$$

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.

$$5x-3y-4=34,-3x+5y-18=34$$

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

$$5x-3y-4=34$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$5x-3y=38$$

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

$$5x=3y+38$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(3y+38\right)$$

Whakareatia \frac{1}{5} ki te 3y+38.

$$x=\frac{3}{5}y+\frac{38}{5}$$

Whakakapia te \frac{3y+38}{5} mō te x ki tērā atu whārite, -3x+5y-18=34.

$$-3\left(\frac{3}{5}y+\frac{38}{5}\right)+5y-18=34$$

Whakareatia -3 ki te \frac{3y+38}{5}.

$$-\frac{9}{5}y-\frac{114}{5}+5y-18=34$$

Tāpiri -\frac{9y}{5} ki te 5y.

$$\frac{16}{5}y-\frac{114}{5}-18=34$$

Tāpiri -\frac{114}{5} ki te -18.

$$\frac{16}{5}y-\frac{204}{5}=34$$

Me tāpiri \frac{204}{5} ki ngā taha e rua o te whārite.

$$\frac{16}{5}y=\frac{374}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{16}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{187}{8}$$

Whakaurua te \frac{187}{8} mō y ki x=\frac{3}{5}y+\frac{38}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{5}\times \frac{187}{8}+\frac{38}{5}$$

Whakareatia \frac{3}{5} ki te \frac{187}{8} 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.

$$x=\frac{561}{40}+\frac{38}{5}$$

Tāpiri \frac{38}{5} ki te \frac{561}{40} 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.

$$x=\frac{173}{8}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{173}{8},y=\frac{187}{8}$$

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.

$$5x-3y-4=34,-3x+5y-18=34$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-3\\-3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}38\\52\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-3\\-3&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-3\\-3&5\end{matrix}\right))\left(\begin{matrix}5&-3\\-3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-3\\-3&5\end{matrix}\right))\left(\begin{matrix}38\\52\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-3\\-3&5\end{matrix}\right))\left(\begin{matrix}38\\52\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-3\\-3&5\end{matrix}\right))\left(\begin{matrix}38\\52\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5\times 5-\left(-3\left(-3\right)\right)}&-\frac{-3}{5\times 5-\left(-3\left(-3\right)\right)}\\-\frac{-3}{5\times 5-\left(-3\left(-3\right)\right)}&\frac{5}{5\times 5-\left(-3\left(-3\right)\right)}\end{matrix}\right)\left(\begin{matrix}38\\52\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{16}&\frac{3}{16}\\\frac{3}{16}&\frac{5}{16}\end{matrix}\right)\left(\begin{matrix}38\\52\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{16}\times 38+\frac{3}{16}\times 52\\\frac{3}{16}\times 38+\frac{5}{16}\times 52\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{173}{8}\\\frac{187}{8}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{173}{8},y=\frac{187}{8}$$

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.

$$5x-3y-4=34,-3x+5y-18=34$$

Kia ōrite ai a 5x me -3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$-3\times 5x-3\left(-3\right)y-3\left(-4\right)=-3\times 34,5\left(-3\right)x+5\times 5y+5\left(-18\right)=5\times 34$$

Whakarūnātia.

$$-15x+9y+12=-102,-15x+25y-90=170$$

Me tango -15x+25y-90=170 mai i -15x+9y+12=-102 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-15x+15x+9y-25y+12+90=-102-170$$

Tāpiri -15x ki te 15x. Ka whakakore atu ngā kupu -15x me 15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$9y-25y+12+90=-102-170$$

Tāpiri 9y ki te -25y.

$$-16y+12+90=-102-170$$

Tāpiri 12 ki te 90.

$$-16y+102=-102-170$$

Tāpiri -102 ki te -170.

$$-16y+102=-272$$

Me tango 102 mai i ngā taha e rua o te whārite.

$$-16y=-374$$

Whakawehea ngā taha e rua ki te -16.

$$y=\frac{187}{8}$$

Whakaurua te \frac{187}{8} mō y ki -3x+5y-18=34. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-3x+5\times \frac{187}{8}-18=34$$

Whakareatia 5 ki te \frac{187}{8}.

$$-3x+\frac{935}{8}-18=34$$

Tāpiri \frac{935}{8} ki te -18.

$$-3x+\frac{791}{8}=34$$

Me tango \frac{791}{8} mai i ngā taha e rua o te whārite.

$$-3x=-\frac{519}{8}$$

Whakawehea ngā taha e rua ki te -3.

$$x=\frac{173}{8}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{173}{8},y=\frac{187}{8}$$