Whakaoti i te {l}{x/3-y/2=8}{x/5+y/3=1}

$$\left. \begin{array} { l } { \frac { x } { 3 } - \frac { y } { 2 } = 8 } \\ { \frac { x } { 5 } + \frac { y } { 3 } = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=15$$

$$y=-6$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2.

$$2x-3y=48$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 15, arā, te tauraro pātahi he tino iti rawa te kitea o 5,3.

$$3x+5y=15$$

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.

$$2x-3y=48,3x+5y=15$$

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

$$2x-3y=48$$

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

$$2x=3y+48$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(3y+48\right)$$

Whakareatia \frac{1}{2} ki te 48+3y.

$$x=\frac{3}{2}y+24$$

Whakakapia te \frac{3y}{2}+24 mō te x ki tērā atu whārite, 3x+5y=15.

$$3\left(\frac{3}{2}y+24\right)+5y=15$$

Whakareatia 3 ki te \frac{3y}{2}+24.

$$\frac{9}{2}y+72+5y=15$$

Tāpiri \frac{9y}{2} ki te 5y.

$$\frac{19}{2}y+72=15$$

Me tango 72 mai i ngā taha e rua o te whārite.

$$\frac{19}{2}y=-57$$

Whakawehea ngā taha e rua o te whārite ki te \frac{19}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-6$$

Whakaurua te -6 mō y ki x=\frac{3}{2}y+24. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{2}\left(-6\right)+24$$

Whakareatia \frac{3}{2} ki te -6.

$$x=-9+24$$

Tāpiri 24 ki te -9.

$$x=15$$

Kua oti te pūnaha te whakatau.

$$x=15,y=-6$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2.

$$2x-3y=48$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 15, arā, te tauraro pātahi he tino iti rawa te kitea o 5,3.

$$3x+5y=15$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$2x-3y=48,3x+5y=15$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-3\\3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}48\\15\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-3\\3&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-3\\3&5\end{matrix}\right))\left(\begin{matrix}2&-3\\3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\3&5\end{matrix}\right))\left(\begin{matrix}48\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\3&5\end{matrix}\right))\left(\begin{matrix}48\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\3&5\end{matrix}\right))\left(\begin{matrix}48\\15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{2\times 5-\left(-3\times 3\right)}&-\frac{-3}{2\times 5-\left(-3\times 3\right)}\\-\frac{3}{2\times 5-\left(-3\times 3\right)}&\frac{2}{2\times 5-\left(-3\times 3\right)}\end{matrix}\right)\left(\begin{matrix}48\\15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{19}&\frac{3}{19}\\-\frac{3}{19}&\frac{2}{19}\end{matrix}\right)\left(\begin{matrix}48\\15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{19}\times 48+\frac{3}{19}\times 15\\-\frac{3}{19}\times 48+\frac{2}{19}\times 15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\-6\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=15,y=-6$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2.

$$2x-3y=48$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 15, arā, te tauraro pātahi he tino iti rawa te kitea o 5,3.

$$3x+5y=15$$

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.

$$2x-3y=48,3x+5y=15$$

Kia ōrite ai a 2x 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 2.

$$3\times 2x+3\left(-3\right)y=3\times 48,2\times 3x+2\times 5y=2\times 15$$

Whakarūnātia.

$$6x-9y=144,6x+10y=30$$

Me tango 6x+10y=30 mai i 6x-9y=144 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-9y-10y=144-30$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-9y-10y=144-30$$

Tāpiri -9y ki te -10y.

$$-19y=144-30$$

Tāpiri 144 ki te -30.

$$-19y=114$$

Whakawehea ngā taha e rua ki te -19.

$$y=-6$$

Whakaurua te -6 mō y ki 3x+5y=15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+5\left(-6\right)=15$$

Whakareatia 5 ki te -6.

$$3x-30=15$$

Me tāpiri 30 ki ngā taha e rua o te whārite.

$$3x=45$$

Whakawehea ngā taha e rua ki te 3.

$$x=15$$

Kua oti te pūnaha te whakatau.

$$x=15,y=-6$$