Whakaoti i te {l}{2x-3y=-1}{5x+2y=26}

$$\left. \begin{array} { l } { 2 x - 3 y = - 1 } \\ { 5 x + 2 y = 26 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=3$$

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.

$$2x-3y=-1,5x+2y=26$$

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

$$2x-3y=-1$$

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

$$2x=3y-1$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(3y-1\right)$$

Whakareatia \frac{1}{2} ki te 3y-1.

$$x=\frac{3}{2}y-\frac{1}{2}$$

Whakakapia te \frac{3y-1}{2} mō te x ki tērā atu whārite, 5x+2y=26.

$$5\left(\frac{3}{2}y-\frac{1}{2}\right)+2y=26$$

Whakareatia 5 ki te \frac{3y-1}{2}.

$$\frac{15}{2}y-\frac{5}{2}+2y=26$$

Tāpiri \frac{15y}{2} ki te 2y.

$$\frac{19}{2}y-\frac{5}{2}=26$$

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

$$\frac{19}{2}y=\frac{57}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{19}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=3$$

Whakaurua te 3 mō y ki x=\frac{3}{2}y-\frac{1}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{2}\times 3-\frac{1}{2}$$

Whakareatia \frac{3}{2} ki te 3.

$$x=\frac{9-1}{2}$$

Tāpiri -\frac{1}{2} ki te \frac{9}{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.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=3$$

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.

$$2x-3y=-1,5x+2y=26$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-3\\5&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\26\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-3\\5&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-3\\5&2\end{matrix}\right))\left(\begin{matrix}2&-3\\5&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\5&2\end{matrix}\right))\left(\begin{matrix}-1\\26\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\5&2\end{matrix}\right))\left(\begin{matrix}-1\\26\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\5&2\end{matrix}\right))\left(\begin{matrix}-1\\26\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2\times 2-\left(-3\times 5\right)}&-\frac{-3}{2\times 2-\left(-3\times 5\right)}\\-\frac{5}{2\times 2-\left(-3\times 5\right)}&\frac{2}{2\times 2-\left(-3\times 5\right)}\end{matrix}\right)\left(\begin{matrix}-1\\26\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{19}&\frac{3}{19}\\-\frac{5}{19}&\frac{2}{19}\end{matrix}\right)\left(\begin{matrix}-1\\26\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{19}\left(-1\right)+\frac{3}{19}\times 26\\-\frac{5}{19}\left(-1\right)+\frac{2}{19}\times 26\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=3$$

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.

$$2x-3y=-1,5x+2y=26$$

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.

$$5\times 2x+5\left(-3\right)y=5\left(-1\right),2\times 5x+2\times 2y=2\times 26$$

Whakarūnātia.

$$10x-15y=-5,10x+4y=52$$

Me tango 10x+4y=52 mai i 10x-15y=-5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x-15y-4y=-5-52$$

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.

$$-15y-4y=-5-52$$

Tāpiri -15y ki te -4y.

$$-19y=-5-52$$

Tāpiri -5 ki te -52.

$$-19y=-57$$

Whakawehea ngā taha e rua ki te -19.

$$y=3$$

Whakaurua te 3 mō y ki 5x+2y=26. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x+2\times 3=26$$

Whakareatia 2 ki te 3.

$$5x+6=26$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$5x=20$$

Whakawehea ngā taha e rua ki te 5.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=3$$