Whakaoti i te {l}{5x+5y=15}{4x+10y=-2}

$$\left\{ \begin{array} { l } { 5 x + 5 y = 15 } \\ { 4 x + 10 y = - 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{16}{3} = 5\frac{1}{3} \approx 5.333333333$$

$$y = -\frac{7}{3} = -2\frac{1}{3} \approx -2.333333333$$

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.

$$5x+5y=15,4x+10y=-2$$

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

$$5x+5y=15$$

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

$$5x=-5y+15$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-5y+15\right)$$

Whakareatia \frac{1}{5} ki te -5y+15.

$$x=-y+3$$

Whakakapia te -y+3 mō te x ki tērā atu whārite, 4x+10y=-2.

$$4\left(-y+3\right)+10y=-2$$

Whakareatia 4 ki te -y+3.

$$-4y+12+10y=-2$$

Tāpiri -4y ki te 10y.

$$6y+12=-2$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$6y=-14$$

Whakawehea ngā taha e rua ki te 6.

$$y=-\frac{7}{3}$$

Whakaurua te -\frac{7}{3} mō y ki x=-y+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\left(-\frac{7}{3}\right)+3$$

Whakareatia -1 ki te -\frac{7}{3}.

$$x=\frac{7}{3}+3$$

Tāpiri 3 ki te \frac{7}{3}.

$$x=\frac{16}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{16}{3},y=-\frac{7}{3}$$

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.

$$5x+5y=15,4x+10y=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&5\\4&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\-2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&5\\4&10\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&5\\4&10\end{matrix}\right))\left(\begin{matrix}5&5\\4&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&5\\4&10\end{matrix}\right))\left(\begin{matrix}15\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&5\\4&10\end{matrix}\right))\left(\begin{matrix}15\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&5\\4&10\end{matrix}\right))\left(\begin{matrix}15\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{5\times 10-5\times 4}&-\frac{5}{5\times 10-5\times 4}\\-\frac{4}{5\times 10-5\times 4}&\frac{5}{5\times 10-5\times 4}\end{matrix}\right)\left(\begin{matrix}15\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&-\frac{1}{6}\\-\frac{2}{15}&\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}15\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 15-\frac{1}{6}\left(-2\right)\\-\frac{2}{15}\times 15+\frac{1}{6}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{16}{3}\\-\frac{7}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{16}{3},y=-\frac{7}{3}$$

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.

$$5x+5y=15,4x+10y=-2$$

Kia ōrite ai a 5x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$4\times 5x+4\times 5y=4\times 15,5\times 4x+5\times 10y=5\left(-2\right)$$

Whakarūnātia.

$$20x+20y=60,20x+50y=-10$$

Me tango 20x+50y=-10 mai i 20x+20y=60 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$20x-20x+20y-50y=60+10$$

Tāpiri 20x ki te -20x. Ka whakakore atu ngā kupu 20x me -20x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$20y-50y=60+10$$

Tāpiri 20y ki te -50y.

$$-30y=60+10$$

Tāpiri 60 ki te 10.

$$-30y=70$$

Whakawehea ngā taha e rua ki te -30.

$$y=-\frac{7}{3}$$

Whakaurua te -\frac{7}{3} mō y ki 4x+10y=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+10\left(-\frac{7}{3}\right)=-2$$

Whakareatia 10 ki te -\frac{7}{3}.

$$4x-\frac{70}{3}=-2$$

Me tāpiri \frac{70}{3} ki ngā taha e rua o te whārite.

$$4x=\frac{64}{3}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{16}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{16}{3},y=-\frac{7}{3}$$