Whakaoti i te {l}{5x+7y=2060}{1020=2060-(2x+4y)}

$$\left\{ \begin{array} { l } { 5 x + 7 y = 2060 } \\ { 1020 = 2060 - ( 2 x + 4 y ) } \end{array} \right.$$

Whakaoti mō x, y

$$x=160$$

$$y=180$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Hei kimi i te tauaro o 2x+4y, kimihia te tauaro o ia taurangi.

$$1020=2060-2x-4y$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$2060-2x-4y=1020$$

Tangohia te 2060 mai i ngā taha e rua.

$$-2x-4y=1020-2060$$

Tangohia te 2060 i te 1020, ka -1040.

$$-2x-4y=-1040$$

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.

$$5x+7y=2060,-2x-4y=-1040$$

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

$$5x+7y=2060$$

Me tango 7y mai i ngā taha e rua o te whārite.

$$5x=-7y+2060$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-7y+2060\right)$$

Whakareatia \frac{1}{5} ki te -7y+2060.

$$x=-\frac{7}{5}y+412$$

Whakakapia te -\frac{7y}{5}+412 mō te x ki tērā atu whārite, -2x-4y=-1040.

$$-2\left(-\frac{7}{5}y+412\right)-4y=-1040$$

Whakareatia -2 ki te -\frac{7y}{5}+412.

$$\frac{14}{5}y-824-4y=-1040$$

Tāpiri \frac{14y}{5} ki te -4y.

$$-\frac{6}{5}y-824=-1040$$

Me tāpiri 824 ki ngā taha e rua o te whārite.

$$-\frac{6}{5}y=-216$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{6}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=180$$

Whakaurua te 180 mō y ki x=-\frac{7}{5}y+412. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{7}{5}\times 180+412$$

Whakareatia -\frac{7}{5} ki te 180.

$$x=-252+412$$

Tāpiri 412 ki te -252.

$$x=160$$

Kua oti te pūnaha te whakatau.

$$x=160,y=180$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Hei kimi i te tauaro o 2x+4y, kimihia te tauaro o ia taurangi.

$$1020=2060-2x-4y$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$2060-2x-4y=1020$$

Tangohia te 2060 mai i ngā taha e rua.

$$-2x-4y=1020-2060$$

Tangohia te 2060 i te 1020, ka -1040.

$$-2x-4y=-1040$$

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

$$5x+7y=2060,-2x-4y=-1040$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&7\\-2&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2060\\-1040\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&7\\-2&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&7\\-2&-4\end{matrix}\right))\left(\begin{matrix}5&7\\-2&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&7\\-2&-4\end{matrix}\right))\left(\begin{matrix}2060\\-1040\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&7\\-2&-4\end{matrix}\right))\left(\begin{matrix}2060\\-1040\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&7\\-2&-4\end{matrix}\right))\left(\begin{matrix}2060\\-1040\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{5\left(-4\right)-7\left(-2\right)}&-\frac{7}{5\left(-4\right)-7\left(-2\right)}\\-\frac{-2}{5\left(-4\right)-7\left(-2\right)}&\frac{5}{5\left(-4\right)-7\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}2060\\-1040\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}&\frac{7}{6}\\-\frac{1}{3}&-\frac{5}{6}\end{matrix}\right)\left(\begin{matrix}2060\\-1040\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}\times 2060+\frac{7}{6}\left(-1040\right)\\-\frac{1}{3}\times 2060-\frac{5}{6}\left(-1040\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}160\\180\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=160,y=180$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Hei kimi i te tauaro o 2x+4y, kimihia te tauaro o ia taurangi.

$$1020=2060-2x-4y$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$2060-2x-4y=1020$$

Tangohia te 2060 mai i ngā taha e rua.

$$-2x-4y=1020-2060$$

Tangohia te 2060 i te 1020, ka -1040.

$$-2x-4y=-1040$$

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.

$$5x+7y=2060,-2x-4y=-1040$$

Kia ōrite ai a 5x me -2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$-2\times 5x-2\times 7y=-2\times 2060,5\left(-2\right)x+5\left(-4\right)y=5\left(-1040\right)$$

Whakarūnātia.

$$-10x-14y=-4120,-10x-20y=-5200$$

Me tango -10x-20y=-5200 mai i -10x-14y=-4120 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-10x+10x-14y+20y=-4120+5200$$

Tāpiri -10x ki te 10x. Ka whakakore atu ngā kupu -10x me 10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-14y+20y=-4120+5200$$

Tāpiri -14y ki te 20y.

$$6y=-4120+5200$$

Tāpiri -4120 ki te 5200.

$$6y=1080$$

Whakawehea ngā taha e rua ki te 6.

$$y=180$$

Whakaurua te 180 mō y ki -2x-4y=-1040. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x-4\times 180=-1040$$

Whakareatia -4 ki te 180.

$$-2x-720=-1040$$

Me tāpiri 720 ki ngā taha e rua o te whārite.

$$-2x=-320$$

Whakawehea ngā taha e rua ki te -2.

$$x=160$$

Kua oti te pūnaha te whakatau.

$$x=160,y=180$$