Whakaoti i te {l}{2x-y=5}{-x=30-3y}

$$\left\{ \begin{array} { l } { 2 x - y = 5 } \\ { - x = 30 - 3 y } \end{array} \right.$$

Whakaoti mō x, y

$$x=9$$

$$y=13$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Me tāpiri te 3y ki ngā taha e rua.

$$-x+3y=30$$

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.

$$2x-y=5,-x+3y=30$$

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

$$2x-y=5$$

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

$$2x=y+5$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(y+5\right)$$

Whakareatia \frac{1}{2} ki te y+5.

$$x=\frac{1}{2}y+\frac{5}{2}$$

Whakakapia te \frac{5+y}{2} mō te x ki tērā atu whārite, -x+3y=30.

$$-\left(\frac{1}{2}y+\frac{5}{2}\right)+3y=30$$

Whakareatia -1 ki te \frac{5+y}{2}.

$$-\frac{1}{2}y-\frac{5}{2}+3y=30$$

Tāpiri -\frac{y}{2} ki te 3y.

$$\frac{5}{2}y-\frac{5}{2}=30$$

Me tāpiri \frac{5}{2} ki ngā taha e rua o te whārite.

$$\frac{5}{2}y=\frac{65}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=13$$

Whakaurua te 13 mō y ki x=\frac{1}{2}y+\frac{5}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{2}\times 13+\frac{5}{2}$$

Whakareatia \frac{1}{2} ki te 13.

$$x=\frac{13+5}{2}$$

Tāpiri \frac{5}{2} ki te \frac{13}{2} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=13$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Me tāpiri te 3y ki ngā taha e rua.

$$-x+3y=30$$

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

$$2x-y=5,-x+3y=30$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-1\\-1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\30\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-1\\-1&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-1\\-1&3\end{matrix}\right))\left(\begin{matrix}2&-1\\-1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\-1&3\end{matrix}\right))\left(\begin{matrix}5\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\-1&3\end{matrix}\right))\left(\begin{matrix}5\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\-1&3\end{matrix}\right))\left(\begin{matrix}5\\30\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2\times 3-\left(-\left(-1\right)\right)}&-\frac{-1}{2\times 3-\left(-\left(-1\right)\right)}\\-\frac{-1}{2\times 3-\left(-\left(-1\right)\right)}&\frac{2}{2\times 3-\left(-\left(-1\right)\right)}\end{matrix}\right)\left(\begin{matrix}5\\30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5}&\frac{1}{5}\\\frac{1}{5}&\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}5\\30\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5}\times 5+\frac{1}{5}\times 30\\\frac{1}{5}\times 5+\frac{2}{5}\times 30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\13\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=9,y=13$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Me tāpiri te 3y ki ngā taha e rua.

$$-x+3y=30$$

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.

$$2x-y=5,-x+3y=30$$

Kia ōrite ai a 2x me -x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$-2x-\left(-y\right)=-5,2\left(-1\right)x+2\times 3y=2\times 30$$

Whakarūnātia.

$$-2x+y=-5,-2x+6y=60$$

Me tango -2x+6y=60 mai i -2x+y=-5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x+y-6y=-5-60$$

Tāpiri -2x ki te 2x. Ka whakakore atu ngā kupu -2x me 2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$y-6y=-5-60$$

Tāpiri y ki te -6y.

$$-5y=-5-60$$

Tāpiri -5 ki te -60.

$$-5y=-65$$

Whakawehea ngā taha e rua ki te -5.

$$y=13$$

Whakaurua te 13 mō y ki -x+3y=30. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-x+3\times 13=30$$

Whakareatia 3 ki te 13.

$$-x+39=30$$

Me tango 39 mai i ngā taha e rua o te whārite.

$$-x=-9$$

Whakawehea ngā taha e rua ki te -1.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=13$$