Whakaoti i te {l}{2x+y=17}{5x-5y=5}

$$\left. \begin{array} { l } { 2 x + y = 17 } \\ { 5 x - 5 y = 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=6$$

$$y=5$$

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.

$$2x+y=17,5x-5y=5$$

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

$$2x+y=17$$

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

$$2x=-y+17$$

Whakawehea ngā taha e rua ki te 2.

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

Whakareatia \frac{1}{2} ki te -y+17.

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

Whakakapia te \frac{-y+17}{2} mō te x ki tērā atu whārite, 5x-5y=5.

$$5\left(-\frac{1}{2}y+\frac{17}{2}\right)-5y=5$$

Whakareatia 5 ki te \frac{-y+17}{2}.

$$-\frac{5}{2}y+\frac{85}{2}-5y=5$$

Tāpiri -\frac{5y}{2} ki te -5y.

$$-\frac{15}{2}y+\frac{85}{2}=5$$

Me tango \frac{85}{2} mai i ngā taha e rua o te whārite.

$$-\frac{15}{2}y=-\frac{75}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{15}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=5$$

Whakaurua te 5 mō y ki x=-\frac{1}{2}y+\frac{17}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times 5+\frac{17}{2}$$

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

$$x=\frac{-5+17}{2}$$

Tāpiri \frac{17}{2} ki te -\frac{5}{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.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=5$$

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.

$$2x+y=17,5x-5y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&1\\5&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}17\\5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&1\\5&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&1\\5&-5\end{matrix}\right))\left(\begin{matrix}2&1\\5&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\5&-5\end{matrix}\right))\left(\begin{matrix}17\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\5&-5\end{matrix}\right))\left(\begin{matrix}17\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\5&-5\end{matrix}\right))\left(\begin{matrix}17\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{2\left(-5\right)-5}&-\frac{1}{2\left(-5\right)-5}\\-\frac{5}{2\left(-5\right)-5}&\frac{2}{2\left(-5\right)-5}\end{matrix}\right)\left(\begin{matrix}17\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&\frac{1}{15}\\\frac{1}{3}&-\frac{2}{15}\end{matrix}\right)\left(\begin{matrix}17\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 17+\frac{1}{15}\times 5\\\frac{1}{3}\times 17-\frac{2}{15}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=6,y=5$$

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.

$$2x+y=17,5x-5y=5$$

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

$$5\times 2x+5y=5\times 17,2\times 5x+2\left(-5\right)y=2\times 5$$

Whakarūnātia.

$$10x+5y=85,10x-10y=10$$

Me tango 10x-10y=10 mai i 10x+5y=85 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x+5y+10y=85-10$$

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.

$$5y+10y=85-10$$

Tāpiri 5y ki te 10y.

$$15y=85-10$$

Tāpiri 85 ki te -10.

$$15y=75$$

Whakawehea ngā taha e rua ki te 15.

$$y=5$$

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

$$5x-5\times 5=5$$

Whakareatia -5 ki te 5.

$$5x-25=5$$

Me tāpiri 25 ki ngā taha e rua o te whārite.

$$5x=30$$

Whakawehea ngā taha e rua ki te 5.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=5$$