Whakaoti i te {l}{x+y=36}{5/7=x/y}

$$\left\{ \begin{array} { l } { x + y = 36 } \\ { \frac { 5 } { 7 } = \frac { x } { y } } \end{array} \right.$$

Whakaoti mō x, y

$$x=15$$

$$y=21$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tē taea kia ōrite te tāupe y ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 7y, arā, te tauraro pātahi he tino iti rawa te kitea o 7,y.

$$5y=7x$$

Tangohia te 7x mai i ngā taha e rua.

$$5y-7x=0$$

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.

$$x+y=36,-7x+5y=0$$

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

$$x+y=36$$

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

$$x=-y+36$$

Whakakapia te -y+36 mō te x ki tērā atu whārite, -7x+5y=0.

$$-7\left(-y+36\right)+5y=0$$

Whakareatia -7 ki te -y+36.

$$7y-252+5y=0$$

Tāpiri 7y ki te 5y.

$$12y-252=0$$

Me tāpiri 252 ki ngā taha e rua o te whārite.

$$12y=252$$

Whakawehea ngā taha e rua ki te 12.

$$y=21$$

Whakaurua te 21 mō y ki x=-y+36. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-21+36$$

Tāpiri 36 ki te -21.

$$x=15$$

Kua oti te pūnaha te whakatau.

$$x=15,y=21$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tē taea kia ōrite te tāupe y ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 7y, arā, te tauraro pātahi he tino iti rawa te kitea o 7,y.

$$5y=7x$$

Tangohia te 7x mai i ngā taha e rua.

$$5y-7x=0$$

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

$$x+y=36,-7x+5y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\-7&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}36\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\-7&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\-7&5\end{matrix}\right))\left(\begin{matrix}1&1\\-7&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-7&5\end{matrix}\right))\left(\begin{matrix}36\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-7&5\end{matrix}\right))\left(\begin{matrix}36\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-7&5\end{matrix}\right))\left(\begin{matrix}36\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5-\left(-7\right)}&-\frac{1}{5-\left(-7\right)}\\-\frac{-7}{5-\left(-7\right)}&\frac{1}{5-\left(-7\right)}\end{matrix}\right)\left(\begin{matrix}36\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{12}&-\frac{1}{12}\\\frac{7}{12}&\frac{1}{12}\end{matrix}\right)\left(\begin{matrix}36\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{12}\times 36\\\frac{7}{12}\times 36\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\21\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=15,y=21$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tē taea kia ōrite te tāupe y ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 7y, arā, te tauraro pātahi he tino iti rawa te kitea o 7,y.

$$5y=7x$$

Tangohia te 7x mai i ngā taha e rua.

$$5y-7x=0$$

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.

$$x+y=36,-7x+5y=0$$

Kia ōrite ai a x me -7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$-7x-7y=-7\times 36,-7x+5y=0$$

Whakarūnātia.

$$-7x-7y=-252,-7x+5y=0$$

Me tango -7x+5y=0 mai i -7x-7y=-252 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-7x+7x-7y-5y=-252$$

Tāpiri -7x ki te 7x. Ka whakakore atu ngā kupu -7x me 7x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-7y-5y=-252$$

Tāpiri -7y ki te -5y.

$$-12y=-252$$

Whakawehea ngā taha e rua ki te -12.

$$y=21$$

Whakaurua te 21 mō y ki -7x+5y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-7x+5\times 21=0$$

Whakareatia 5 ki te 21.

$$-7x+105=0$$

Me tango 105 mai i ngā taha e rua o te whārite.

$$-7x=-105$$

Whakawehea ngā taha e rua ki te -7.

$$x=15$$

Kua oti te pūnaha te whakatau.

$$x=15,y=21$$