Whakaoti i te {l}{2.5x+2.5y=17}{-1.5x-7.5y=-33}

$$\left. \begin{array} { l } { 2.5 x + 2.5 y = 17 } \\ { - 1.5 x - 7.5 y = - 33 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=3.8$$

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.

$$2.5x+2.5y=17,-1.5x-7.5y=-33$$

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

$$2.5x+2.5y=17$$

Me tango \frac{5y}{2} mai i ngā taha e rua o te whārite.

$$2.5x=-2.5y+17$$

Whakawehea ngā taha e rua o te whārite ki te 2.5, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=0.4\left(-2.5y+17\right)$$

Whakareatia 0.4 ki te -\frac{5y}{2}+17.

$$x=-y+6.8$$

Whakakapia te -y+6.8 mō te x ki tērā atu whārite, -1.5x-7.5y=-33.

$$-1.5\left(-y+6.8\right)-7.5y=-33$$

Whakareatia -1.5 ki te -y+6.8.

$$1.5y-10.2-7.5y=-33$$

Tāpiri \frac{3y}{2} ki te -\frac{15y}{2}.

$$-6y-10.2=-33$$

Me tāpiri 10.2 ki ngā taha e rua o te whārite.

$$-6y=-22.8$$

Whakawehea ngā taha e rua ki te -6.

$$y=3.8$$

Whakaurua te 3.8 mō y ki x=-y+6.8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-3.8+6.8$$

Whakareatia -1 ki te 3.8.

$$x=\frac{-19+34}{5}$$

Tāpiri 6.8 ki te -3.8 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.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=3.8$$

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.

$$2.5x+2.5y=17,-1.5x-7.5y=-33$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2.5&2.5\\-1.5&-7.5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}17\\-33\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2.5&2.5\\-1.5&-7.5\end{matrix}\right).

$$inverse(\left(\begin{matrix}2.5&2.5\\-1.5&-7.5\end{matrix}\right))\left(\begin{matrix}2.5&2.5\\-1.5&-7.5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2.5&2.5\\-1.5&-7.5\end{matrix}\right))\left(\begin{matrix}17\\-33\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2.5&2.5\\-1.5&-7.5\end{matrix}\right))\left(\begin{matrix}17\\-33\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2.5&2.5\\-1.5&-7.5\end{matrix}\right))\left(\begin{matrix}17\\-33\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7.5}{2.5\left(-7.5\right)-2.5\left(-1.5\right)}&-\frac{2.5}{2.5\left(-7.5\right)-2.5\left(-1.5\right)}\\-\frac{-1.5}{2.5\left(-7.5\right)-2.5\left(-1.5\right)}&\frac{2.5}{2.5\left(-7.5\right)-2.5\left(-1.5\right)}\end{matrix}\right)\left(\begin{matrix}17\\-33\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{6}\\-\frac{1}{10}&-\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}17\\-33\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 17+\frac{1}{6}\left(-33\right)\\-\frac{1}{10}\times 17-\frac{1}{6}\left(-33\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\\frac{19}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=\frac{19}{5}$$

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.

$$2.5x+2.5y=17,-1.5x-7.5y=-33$$

Kia ōrite ai a \frac{5x}{2} me -\frac{3x}{2}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -1.5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.5.

$$-1.5\times 2.5x-1.5\times 2.5y=-1.5\times 17,2.5\left(-1.5\right)x+2.5\left(-7.5\right)y=2.5\left(-33\right)$$

Whakarūnātia.

$$-3.75x-3.75y=-25.5,-3.75x-18.75y=-82.5$$

Me tango -3.75x-18.75y=-82.5 mai i -3.75x-3.75y=-25.5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-3.75x+3.75x-3.75y+18.75y=\frac{-51+165}{2}$$

Tāpiri -\frac{15x}{4} ki te \frac{15x}{4}. Ka whakakore atu ngā kupu -\frac{15x}{4} me \frac{15x}{4}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-3.75y+18.75y=\frac{-51+165}{2}$$

Tāpiri -\frac{15y}{4} ki te \frac{75y}{4}.

$$15y=\frac{-51+165}{2}$$

Tāpiri -25.5 ki te 82.5 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.

$$15y=57$$

Whakawehea ngā taha e rua ki te 15.

$$y=\frac{19}{5}$$

Whakaurua te \frac{19}{5} mō y ki -1.5x-7.5y=-33. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-1.5x-7.5\times \frac{19}{5}=-33$$

Whakareatia -7.5 ki te \frac{19}{5} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$-1.5x-\frac{57}{2}=-33$$

Me tāpiri \frac{57}{2} ki ngā taha e rua o te whārite.

$$-1.5x=-\frac{9}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -1.5, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=\frac{19}{5}$$