Whakaoti i te {l}{x/3-y/2=4}{x+y/12=2}

$$\left\{ \begin{array} { l } { \frac { x } { 3 } - \frac { y } { 2 } = 4 } \\ { x + \frac { y } { 12 } = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{48}{19} = 2\frac{10}{19} \approx 2.526315789$$

$$y = -\frac{120}{19} = -6\frac{6}{19} \approx -6.315789474$$

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.

$$2x-3y=24$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 12.

$$12x+y=24$$

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.

$$2x-3y=24,12x+y=24$$

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

$$2x-3y=24$$

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

$$2x=3y+24$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(3y+24\right)$$

Whakareatia \frac{1}{2} ki te 24+3y.

$$x=\frac{3}{2}y+12$$

Whakakapia te \frac{3y}{2}+12 mō te x ki tērā atu whārite, 12x+y=24.

$$12\left(\frac{3}{2}y+12\right)+y=24$$

Whakareatia 12 ki te \frac{3y}{2}+12.

$$18y+144+y=24$$

Tāpiri 18y ki te y.

$$19y+144=24$$

Me tango 144 mai i ngā taha e rua o te whārite.

$$19y=-120$$

Whakawehea ngā taha e rua ki te 19.

$$y=-\frac{120}{19}$$

Whakaurua te -\frac{120}{19} mō y ki x=\frac{3}{2}y+12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{2}\left(-\frac{120}{19}\right)+12$$

Whakareatia \frac{3}{2} ki te -\frac{120}{19} 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.

$$x=-\frac{180}{19}+12$$

Tāpiri 12 ki te -\frac{180}{19}.

$$x=\frac{48}{19}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{48}{19},y=-\frac{120}{19}$$

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.

$$2x-3y=24$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 12.

$$12x+y=24$$

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

$$2x-3y=24,12x+y=24$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-3\\12&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}24\\24\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-3\\12&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-3\\12&1\end{matrix}\right))\left(\begin{matrix}2&-3\\12&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\12&1\end{matrix}\right))\left(\begin{matrix}24\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\12&1\end{matrix}\right))\left(\begin{matrix}24\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\12&1\end{matrix}\right))\left(\begin{matrix}24\\24\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2-\left(-3\times 12\right)}&-\frac{-3}{2-\left(-3\times 12\right)}\\-\frac{12}{2-\left(-3\times 12\right)}&\frac{2}{2-\left(-3\times 12\right)}\end{matrix}\right)\left(\begin{matrix}24\\24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{38}&\frac{3}{38}\\-\frac{6}{19}&\frac{1}{19}\end{matrix}\right)\left(\begin{matrix}24\\24\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{38}\times 24+\frac{3}{38}\times 24\\-\frac{6}{19}\times 24+\frac{1}{19}\times 24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{48}{19}\\-\frac{120}{19}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{48}{19},y=-\frac{120}{19}$$

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.

$$2x-3y=24$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 12.

$$12x+y=24$$

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.

$$2x-3y=24,12x+y=24$$

Kia ōrite ai a 2x me 12x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 12 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$12\times 2x+12\left(-3\right)y=12\times 24,2\times 12x+2y=2\times 24$$

Whakarūnātia.

$$24x-36y=288,24x+2y=48$$

Me tango 24x+2y=48 mai i 24x-36y=288 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$24x-24x-36y-2y=288-48$$

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.

$$-36y-2y=288-48$$

Tāpiri -36y ki te -2y.

$$-38y=288-48$$

Tāpiri 288 ki te -48.

$$-38y=240$$

Whakawehea ngā taha e rua ki te -38.

$$y=-\frac{120}{19}$$

Whakaurua te -\frac{120}{19} mō y ki 12x+y=24. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$12x-\frac{120}{19}=24$$

Me tāpiri \frac{120}{19} ki ngā taha e rua o te whārite.

$$12x=\frac{576}{19}$$

Whakawehea ngā taha e rua ki te 12.

$$x=\frac{48}{19}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{48}{19},y=-\frac{120}{19}$$