Whakaoti i te {l}{2x+3y=5}{7x=4y-5}

$$\left. \begin{array} { l } { 2 x + 3 y = 5 } \\ { 7 x = 4 y - 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{5}{29}\approx 0.172413793$$

$$y = \frac{45}{29} = 1\frac{16}{29} \approx 1.551724138$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 4y mai i ngā taha e rua.

$$7x-4y=-5$$

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.

$$2x+3y=5,7x-4y=-5$$

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

$$2x+3y=5$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$2x=-3y+5$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

Whakakapia te \frac{-3y+5}{2} mō te x ki tērā atu whārite, 7x-4y=-5.

$$7\left(-\frac{3}{2}y+\frac{5}{2}\right)-4y=-5$$

Whakareatia 7 ki te \frac{-3y+5}{2}.

$$-\frac{21}{2}y+\frac{35}{2}-4y=-5$$

Tāpiri -\frac{21y}{2} ki te -4y.

$$-\frac{29}{2}y+\frac{35}{2}=-5$$

Me tango \frac{35}{2} mai i ngā taha e rua o te whārite.

$$-\frac{29}{2}y=-\frac{45}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{29}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{45}{29}$$

Whakaurua te \frac{45}{29} mō y ki x=-\frac{3}{2}y+\frac{5}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times \frac{45}{29}+\frac{5}{2}$$

Whakareatia -\frac{3}{2} ki te \frac{45}{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.

$$x=-\frac{135}{58}+\frac{5}{2}$$

Tāpiri \frac{5}{2} ki te -\frac{135}{58} 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.

$$x=\frac{5}{29}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{29},y=\frac{45}{29}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 4y mai i ngā taha e rua.

$$7x-4y=-5$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$2x+3y=5,7x-4y=-5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\7&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\-5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&3\\7&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\7&-4\end{matrix}\right))\left(\begin{matrix}2&3\\7&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\7&-4\end{matrix}\right))\left(\begin{matrix}5\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\7&-4\end{matrix}\right))\left(\begin{matrix}5\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\7&-4\end{matrix}\right))\left(\begin{matrix}5\\-5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{2\left(-4\right)-3\times 7}&-\frac{3}{2\left(-4\right)-3\times 7}\\-\frac{7}{2\left(-4\right)-3\times 7}&\frac{2}{2\left(-4\right)-3\times 7}\end{matrix}\right)\left(\begin{matrix}5\\-5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{29}&\frac{3}{29}\\\frac{7}{29}&-\frac{2}{29}\end{matrix}\right)\left(\begin{matrix}5\\-5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{29}\times 5+\frac{3}{29}\left(-5\right)\\\frac{7}{29}\times 5-\frac{2}{29}\left(-5\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{29}\\\frac{45}{29}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{5}{29},y=\frac{45}{29}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 4y mai i ngā taha e rua.

$$7x-4y=-5$$

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.

$$2x+3y=5,7x-4y=-5$$

Kia ōrite ai a 2x me 7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$7\times 2x+7\times 3y=7\times 5,2\times 7x+2\left(-4\right)y=2\left(-5\right)$$

Whakarūnātia.

$$14x+21y=35,14x-8y=-10$$

Me tango 14x-8y=-10 mai i 14x+21y=35 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$14x-14x+21y+8y=35+10$$

Tāpiri 14x ki te -14x. Ka whakakore atu ngā kupu 14x me -14x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$21y+8y=35+10$$

Tāpiri 21y ki te 8y.

$$29y=35+10$$

Tāpiri 35 ki te 10.

$$29y=45$$

Whakawehea ngā taha e rua ki te 29.

$$y=\frac{45}{29}$$

Whakaurua te \frac{45}{29} mō y ki 7x-4y=-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x-4\times \frac{45}{29}=-5$$

Whakareatia -4 ki te \frac{45}{29}.

$$7x-\frac{180}{29}=-5$$

Me tāpiri \frac{180}{29} ki ngā taha e rua o te whārite.

$$7x=\frac{35}{29}$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{5}{29}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{5}{29},y=\frac{45}{29}$$