Whakaoti i te {l}{y+2z/3=4}{5y/6+7z/3=8}

$$\left. \begin{array} { l } { \frac { y + 2 z } { 3 } = 4 } \\ { \frac { 5 y } { 6 } + \frac { 7 z } { 3 } = 8 } \end{array} \right.$$

Whakaoti mō y, z

$$y=18$$

$$z=-3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua ki te 3.

$$y+2z=4\times 3$$

Whakareatia te 4 ki te 3, ka 12.

$$y+2z=12$$

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

$$5y+2\times 7z=48$$

Whakareatia te 2 ki te 7, ka 14.

$$5y+14z=48$$

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.

$$y+2z=12,5y+14z=48$$

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

$$y+2z=12$$

Me tango 2z mai i ngā taha e rua o te whārite.

$$y=-2z+12$$

Whakakapia te -2z+12 mō te y ki tērā atu whārite, 5y+14z=48.

$$5\left(-2z+12\right)+14z=48$$

Whakareatia 5 ki te -2z+12.

$$-10z+60+14z=48$$

Tāpiri -10z ki te 14z.

$$4z+60=48$$

Me tango 60 mai i ngā taha e rua o te whārite.

$$4z=-12$$

Whakawehea ngā taha e rua ki te 4.

$$z=-3$$

Whakaurua te -3 mō z ki y=-2z+12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-2\left(-3\right)+12$$

Whakareatia -2 ki te -3.

$$y=6+12$$

Tāpiri 12 ki te 6.

$$y=18$$

Kua oti te pūnaha te whakatau.

$$y=18,z=-3$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua ki te 3.

$$y+2z=4\times 3$$

Whakareatia te 4 ki te 3, ka 12.

$$y+2z=12$$

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

$$5y+2\times 7z=48$$

Whakareatia te 2 ki te 7, ka 14.

$$5y+14z=48$$

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

$$y+2z=12,5y+14z=48$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\5&14\end{matrix}\right)\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}12\\48\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&2\\5&14\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&2\\5&14\end{matrix}\right))\left(\begin{matrix}1&2\\5&14\end{matrix}\right)\left(\begin{matrix}y\\z\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\5&14\end{matrix}\right))\left(\begin{matrix}12\\48\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\z\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\5&14\end{matrix}\right))\left(\begin{matrix}12\\48\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\z\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\5&14\end{matrix}\right))\left(\begin{matrix}12\\48\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}\frac{14}{14-2\times 5}&-\frac{2}{14-2\times 5}\\-\frac{5}{14-2\times 5}&\frac{1}{14-2\times 5}\end{matrix}\right)\left(\begin{matrix}12\\48\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}\frac{7}{2}&-\frac{1}{2}\\-\frac{5}{4}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}12\\48\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}\frac{7}{2}\times 12-\frac{1}{2}\times 48\\-\frac{5}{4}\times 12+\frac{1}{4}\times 48\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}18\\-3\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me z.

$$y=18,z=-3$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua ki te 3.

$$y+2z=4\times 3$$

Whakareatia te 4 ki te 3, ka 12.

$$y+2z=12$$

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

$$5y+2\times 7z=48$$

Whakareatia te 2 ki te 7, ka 14.

$$5y+14z=48$$

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.

$$y+2z=12,5y+14z=48$$

Kia ōrite ai a y me 5y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$5y+5\times 2z=5\times 12,5y+14z=48$$

Whakarūnātia.

$$5y+10z=60,5y+14z=48$$

Me tango 5y+14z=48 mai i 5y+10z=60 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5y-5y+10z-14z=60-48$$

Tāpiri 5y ki te -5y. Ka whakakore atu ngā kupu 5y me -5y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$10z-14z=60-48$$

Tāpiri 10z ki te -14z.

$$-4z=60-48$$

Tāpiri 60 ki te -48.

$$-4z=12$$

Whakawehea ngā taha e rua ki te -4.

$$z=-3$$

Whakaurua te -3 mō z ki 5y+14z=48. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$5y+14\left(-3\right)=48$$

Whakareatia 14 ki te -3.

$$5y-42=48$$

Me tāpiri 42 ki ngā taha e rua o te whārite.

$$5y=90$$

Whakawehea ngā taha e rua ki te 5.

$$y=18$$

Kua oti te pūnaha te whakatau.

$$y=18,z=-3$$