Whakaoti i te {l}{x/8-y=-5/2}{3x+y/3=13}

$$\left. \begin{array} { l } { \frac { x } { 8 } - y = - \frac { 5 } { 2 } } \\ { 3 x + \frac { y } { 3 } = 13 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=3$$

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.

$$\frac{1}{8}x-y=-\frac{5}{2},3x+\frac{1}{3}y=13$$

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

$$\frac{1}{8}x-y=-\frac{5}{2}$$

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

$$\frac{1}{8}x=y-\frac{5}{2}$$

Me whakarea ngā taha e rua ki te 8.

$$x=8\left(y-\frac{5}{2}\right)$$

Whakareatia 8 ki te y-\frac{5}{2}.

$$x=8y-20$$

Whakakapia te 8y-20 mō te x ki tērā atu whārite, 3x+\frac{1}{3}y=13.

$$3\left(8y-20\right)+\frac{1}{3}y=13$$

Whakareatia 3 ki te 8y-20.

$$24y-60+\frac{1}{3}y=13$$

Tāpiri 24y ki te \frac{y}{3}.

$$\frac{73}{3}y-60=13$$

Me tāpiri 60 ki ngā taha e rua o te whārite.

$$\frac{73}{3}y=73$$

Whakawehea ngā taha e rua o te whārite ki te \frac{73}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=3$$

Whakaurua te 3 mō y ki x=8y-20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=8\times 3-20$$

Whakareatia 8 ki te 3.

$$x=24-20$$

Tāpiri -20 ki te 24.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=3$$

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.

$$\frac{1}{8}x-y=-\frac{5}{2},3x+\frac{1}{3}y=13$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}\frac{1}{8}&-1\\3&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{2}\\13\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}\frac{1}{8}&-1\\3&\frac{1}{3}\end{matrix}\right).

$$inverse(\left(\begin{matrix}\frac{1}{8}&-1\\3&\frac{1}{3}\end{matrix}\right))\left(\begin{matrix}\frac{1}{8}&-1\\3&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{8}&-1\\3&\frac{1}{3}\end{matrix}\right))\left(\begin{matrix}-\frac{5}{2}\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{8}&-1\\3&\frac{1}{3}\end{matrix}\right))\left(\begin{matrix}-\frac{5}{2}\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{8}&-1\\3&\frac{1}{3}\end{matrix}\right))\left(\begin{matrix}-\frac{5}{2}\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{1}{3}}{\frac{1}{8}\times \frac{1}{3}-\left(-3\right)}&-\frac{-1}{\frac{1}{8}\times \frac{1}{3}-\left(-3\right)}\\-\frac{3}{\frac{1}{8}\times \frac{1}{3}-\left(-3\right)}&\frac{\frac{1}{8}}{\frac{1}{8}\times \frac{1}{3}-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}-\frac{5}{2}\\13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{73}&\frac{24}{73}\\-\frac{72}{73}&\frac{3}{73}\end{matrix}\right)\left(\begin{matrix}-\frac{5}{2}\\13\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{73}\left(-\frac{5}{2}\right)+\frac{24}{73}\times 13\\-\frac{72}{73}\left(-\frac{5}{2}\right)+\frac{3}{73}\times 13\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=3$$

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.

$$\frac{1}{8}x-y=-\frac{5}{2},3x+\frac{1}{3}y=13$$

Kia ōrite ai a \frac{x}{8} 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 \frac{1}{8}.

$$3\times \frac{1}{8}x+3\left(-1\right)y=3\left(-\frac{5}{2}\right),\frac{1}{8}\times 3x+\frac{1}{8}\times \frac{1}{3}y=\frac{1}{8}\times 13$$

Whakarūnātia.

$$\frac{3}{8}x-3y=-\frac{15}{2},\frac{3}{8}x+\frac{1}{24}y=\frac{13}{8}$$

Me tango \frac{3}{8}x+\frac{1}{24}y=\frac{13}{8} mai i \frac{3}{8}x-3y=-\frac{15}{2} mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{3}{8}x-\frac{3}{8}x-3y-\frac{1}{24}y=-\frac{15}{2}-\frac{13}{8}$$

Tāpiri \frac{3x}{8} ki te -\frac{3x}{8}. Ka whakakore atu ngā kupu \frac{3x}{8} me -\frac{3x}{8}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-3y-\frac{1}{24}y=-\frac{15}{2}-\frac{13}{8}$$

Tāpiri -3y ki te -\frac{y}{24}.

$$-\frac{73}{24}y=-\frac{15}{2}-\frac{13}{8}$$

Tāpiri -\frac{15}{2} ki te -\frac{13}{8} 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.

$$-\frac{73}{24}y=-\frac{73}{8}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{73}{24}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=3$$

Whakaurua te 3 mō y ki 3x+\frac{1}{3}y=13. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+\frac{1}{3}\times 3=13$$

Whakareatia \frac{1}{3} ki te 3.

$$3x+1=13$$

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

$$3x=12$$

Whakawehea ngā taha e rua ki te 3.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=3$$