Whakaoti i te 7x-15y-2=0text{ztal}x+2y=3

$$7 x - 15 y - 2 = 0 \text { ztal } x + 2 y = 3$$

Whakaoti mō x, y

$$x = \frac{49}{29} = 1\frac{20}{29} \approx 1.689655172$$

$$y=\frac{19}{29}\approx 0.655172414$$

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.

$$7x-15y-2=0,x+2y=3$$

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

$$7x-15y-2=0$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$7x-15y=2$$

Me tāpiri 15y ki ngā taha e rua o te whārite.

$$7x=15y+2$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(15y+2\right)$$

Whakareatia \frac{1}{7} ki te 15y+2.

$$x=\frac{15}{7}y+\frac{2}{7}$$

Whakakapia te \frac{15y+2}{7} mō te x ki tērā atu whārite, x+2y=3.

$$\frac{15}{7}y+\frac{2}{7}+2y=3$$

Tāpiri \frac{15y}{7} ki te 2y.

$$\frac{29}{7}y+\frac{2}{7}=3$$

Me tango \frac{2}{7} mai i ngā taha e rua o te whārite.

$$\frac{29}{7}y=\frac{19}{7}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{29}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{19}{29}$$

Whakaurua te \frac{19}{29} mō y ki x=\frac{15}{7}y+\frac{2}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{15}{7}\times \frac{19}{29}+\frac{2}{7}$$

Whakareatia \frac{15}{7} ki te \frac{19}{29} 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.

$$x=\frac{285}{203}+\frac{2}{7}$$

Tāpiri \frac{2}{7} ki te \frac{285}{203} 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.

$$x=\frac{49}{29}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{49}{29},y=\frac{19}{29}$$

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.

$$7x-15y-2=0,x+2y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&-15\\1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&-15\\1&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&-15\\1&2\end{matrix}\right))\left(\begin{matrix}7&-15\\1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-15\\1&2\end{matrix}\right))\left(\begin{matrix}2\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-15\\1&2\end{matrix}\right))\left(\begin{matrix}2\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-15\\1&2\end{matrix}\right))\left(\begin{matrix}2\\3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7\times 2-\left(-15\right)}&-\frac{-15}{7\times 2-\left(-15\right)}\\-\frac{1}{7\times 2-\left(-15\right)}&\frac{7}{7\times 2-\left(-15\right)}\end{matrix}\right)\left(\begin{matrix}2\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{29}&\frac{15}{29}\\-\frac{1}{29}&\frac{7}{29}\end{matrix}\right)\left(\begin{matrix}2\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{29}\times 2+\frac{15}{29}\times 3\\-\frac{1}{29}\times 2+\frac{7}{29}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{49}{29}\\\frac{19}{29}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{49}{29},y=\frac{19}{29}$$

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.

$$7x-15y-2=0,x+2y=3$$

Kia ōrite ai a 7x 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 7.

$$7x-15y-2=0,7x+7\times 2y=7\times 3$$

Whakarūnātia.

$$7x-15y-2=0,7x+14y=21$$

Me tango 7x+14y=21 mai i 7x-15y-2=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$7x-7x-15y-14y-2=-21$$

Tāpiri 7x ki te -7x. Ka whakakore atu ngā kupu 7x me -7x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-15y-14y-2=-21$$

Tāpiri -15y ki te -14y.

$$-29y-2=-21$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$-29y=-19$$

Whakawehea ngā taha e rua ki te -29.

$$y=\frac{19}{29}$$

Whakaurua te \frac{19}{29} mō y ki x+2y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+2\times \frac{19}{29}=3$$

Whakareatia 2 ki te \frac{19}{29}.

$$x+\frac{38}{29}=3$$

Me tango \frac{38}{29} mai i ngā taha e rua o te whārite.

$$x=\frac{49}{29}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{49}{29},y=\frac{19}{29}$$