Whakaoti i te {l}{10x+5y=170}{6x+10y=200}

$$\left\{ \begin{array} { l } { 10 x + 5 y = 170 } \\ { 6 x + 10 y = 200 } \end{array} \right.$$

Whakaoti mō x, y

$$x=10$$

$$y=14$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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.

$$10x+5y=170,6x+10y=200$$

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

$$10x+5y=170$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$10x=-5y+170$$

Whakawehea ngā taha e rua ki te 10.

$$x=\frac{1}{10}\left(-5y+170\right)$$

Whakareatia \frac{1}{10} ki te -5y+170.

$$x=-\frac{1}{2}y+17$$

Whakakapia te -\frac{y}{2}+17 mō te x ki tērā atu whārite, 6x+10y=200.

$$6\left(-\frac{1}{2}y+17\right)+10y=200$$

Whakareatia 6 ki te -\frac{y}{2}+17.

$$-3y+102+10y=200$$

Tāpiri -3y ki te 10y.

$$7y+102=200$$

Me tango 102 mai i ngā taha e rua o te whārite.

$$7y=98$$

Whakawehea ngā taha e rua ki te 7.

$$y=14$$

Whakaurua te 14 mō y ki x=-\frac{1}{2}y+17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times 14+17$$

Whakareatia -\frac{1}{2} ki te 14.

$$x=-7+17$$

Tāpiri 17 ki te -7.

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=14$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$10x+5y=170,6x+10y=200$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}10&5\\6&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}170\\200\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}10&5\\6&10\end{matrix}\right).

$$inverse(\left(\begin{matrix}10&5\\6&10\end{matrix}\right))\left(\begin{matrix}10&5\\6&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&5\\6&10\end{matrix}\right))\left(\begin{matrix}170\\200\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&5\\6&10\end{matrix}\right))\left(\begin{matrix}170\\200\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&5\\6&10\end{matrix}\right))\left(\begin{matrix}170\\200\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{10\times 10-5\times 6}&-\frac{5}{10\times 10-5\times 6}\\-\frac{6}{10\times 10-5\times 6}&\frac{10}{10\times 10-5\times 6}\end{matrix}\right)\left(\begin{matrix}170\\200\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}&-\frac{1}{14}\\-\frac{3}{35}&\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}170\\200\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}\times 170-\frac{1}{14}\times 200\\-\frac{3}{35}\times 170+\frac{1}{7}\times 200\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\14\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=10,y=14$$

Ngā Upane e Whakamahi i te Tangohanga

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.

$$10x+5y=170,6x+10y=200$$

Kia ōrite ai a 10x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 10.

$$6\times 10x+6\times 5y=6\times 170,10\times 6x+10\times 10y=10\times 200$$

Whakarūnātia.

$$60x+30y=1020,60x+100y=2000$$

Me tango 60x+100y=2000 mai i 60x+30y=1020 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$60x-60x+30y-100y=1020-2000$$

Tāpiri 60x ki te -60x. Ka whakakore atu ngā kupu 60x me -60x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$30y-100y=1020-2000$$

Tāpiri 30y ki te -100y.

$$-70y=1020-2000$$

Tāpiri 1020 ki te -2000.

$$-70y=-980$$

Whakawehea ngā taha e rua ki te -70.

$$y=14$$

Whakaurua te 14 mō y ki 6x+10y=200. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x+10\times 14=200$$

Whakareatia 10 ki te 14.

$$6x+140=200$$

Me tango 140 mai i ngā taha e rua o te whārite.

$$6x=60$$

Whakawehea ngā taha e rua ki te 6.

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=14$$