Whakaoti i te 2x-3y+10=0quadtext{KaHa}5x-y+4=0

$$2 x - 3 y + 10 = 0 \quad \text { KaHa } 5 x - y + 4 = 0$$

Whakaoti mō x, y

$$x=-\frac{2}{13}\approx -0.153846154$$

$$y = \frac{42}{13} = 3\frac{3}{13} \approx 3.230769231$$

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.

$$2x-3y+10=0,5x-y+4=0$$

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

$$2x-3y+10=0$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$2x-3y=-10$$

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

$$2x=3y-10$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(3y-10\right)$$

Whakareatia \frac{1}{2} ki te 3y-10.

$$x=\frac{3}{2}y-5$$

Whakakapia te \frac{3y}{2}-5 mō te x ki tērā atu whārite, 5x-y+4=0.

$$5\left(\frac{3}{2}y-5\right)-y+4=0$$

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

$$\frac{15}{2}y-25-y+4=0$$

Tāpiri \frac{15y}{2} ki te -y.

$$\frac{13}{2}y-25+4=0$$

Tāpiri -25 ki te 4.

$$\frac{13}{2}y-21=0$$

Me tāpiri 21 ki ngā taha e rua o te whārite.

$$\frac{13}{2}y=21$$

Whakawehea ngā taha e rua o te whārite ki te \frac{13}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{42}{13}$$

Whakaurua te \frac{42}{13} mō y ki x=\frac{3}{2}y-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{2}\times \frac{42}{13}-5$$

Whakareatia \frac{3}{2} ki te \frac{42}{13} 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.

$$x=\frac{63}{13}-5$$

Tāpiri -5 ki te \frac{63}{13}.

$$x=-\frac{2}{13}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{2}{13},y=\frac{42}{13}$$

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.

$$2x-3y+10=0,5x-y+4=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-3\\5&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-10\\-4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-3\\5&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-3\\5&-1\end{matrix}\right))\left(\begin{matrix}2&-3\\5&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\5&-1\end{matrix}\right))\left(\begin{matrix}-10\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\5&-1\end{matrix}\right))\left(\begin{matrix}-10\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\5&-1\end{matrix}\right))\left(\begin{matrix}-10\\-4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2\left(-1\right)-\left(-3\times 5\right)}&-\frac{-3}{2\left(-1\right)-\left(-3\times 5\right)}\\-\frac{5}{2\left(-1\right)-\left(-3\times 5\right)}&\frac{2}{2\left(-1\right)-\left(-3\times 5\right)}\end{matrix}\right)\left(\begin{matrix}-10\\-4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{13}&\frac{3}{13}\\-\frac{5}{13}&\frac{2}{13}\end{matrix}\right)\left(\begin{matrix}-10\\-4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{13}\left(-10\right)+\frac{3}{13}\left(-4\right)\\-\frac{5}{13}\left(-10\right)+\frac{2}{13}\left(-4\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{13}\\\frac{42}{13}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{2}{13},y=\frac{42}{13}$$

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.

$$2x-3y+10=0,5x-y+4=0$$

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.

$$5\times 2x+5\left(-3\right)y+5\times 10=0,2\times 5x+2\left(-1\right)y+2\times 4=0$$

Whakarūnātia.

$$10x-15y+50=0,10x-2y+8=0$$

Me tango 10x-2y+8=0 mai i 10x-15y+50=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x-15y+2y+50-8=0$$

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.

$$-15y+2y+50-8=0$$

Tāpiri -15y ki te 2y.

$$-13y+50-8=0$$

Tāpiri 50 ki te -8.

$$-13y+42=0$$

Me tango 42 mai i ngā taha e rua o te whārite.

$$-13y=-42$$

Whakawehea ngā taha e rua ki te -13.

$$y=\frac{42}{13}$$

Whakaurua te \frac{42}{13} mō y ki 5x-y+4=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-\frac{42}{13}+4=0$$

Tāpiri -\frac{42}{13} ki te 4.

$$5x+\frac{10}{13}=0$$

Me tango \frac{10}{13} mai i ngā taha e rua o te whārite.

$$5x=-\frac{10}{13}$$

Whakawehea ngā taha e rua ki te 5.

$$x=-\frac{2}{13}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{2}{13},y=\frac{42}{13}$$