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

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

Whakaoti mō x, y

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$y=-\frac{1}{2}=-0.5$$

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.

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

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

$$3x-5y=7$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$3x=5y+7$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(5y+7\right)$$

Whakareatia \frac{1}{3} ki te 5y+7.

$$x=\frac{5}{3}y+\frac{7}{3}$$

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

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

Whakareatia 4 ki te \frac{5y+7}{3}.

$$\frac{20}{3}y+\frac{28}{3}+2y=5$$

Tāpiri \frac{20y}{3} ki te 2y.

$$\frac{26}{3}y+\frac{28}{3}=5$$

Me tango \frac{28}{3} mai i ngā taha e rua o te whārite.

$$\frac{26}{3}y=-\frac{13}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{26}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

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

Whakaurua te -\frac{1}{2} mō y ki x=\frac{5}{3}y+\frac{7}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{3}\left(-\frac{1}{2}\right)+\frac{7}{3}$$

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

$$x=-\frac{5}{6}+\frac{7}{3}$$

Tāpiri \frac{7}{3} ki te -\frac{5}{6} 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.

$$x=\frac{3}{2}$$

Kua oti te pūnaha te whakatau.

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

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-5\\4&2\end{matrix}\right).

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

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

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

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

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

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{13}&\frac{5}{26}\\-\frac{2}{13}&\frac{3}{26}\end{matrix}\right)\left(\begin{matrix}7\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{13}\times 7+\frac{5}{26}\times 5\\-\frac{2}{13}\times 7+\frac{3}{26}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

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

Kia ōrite ai a 3x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

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

Whakarūnātia.

$$12x-20y=28,12x+6y=15$$

Me tango 12x+6y=15 mai i 12x-20y=28 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x-20y-6y=28-15$$

Tāpiri 12x ki te -12x. Ka whakakore atu ngā kupu 12x me -12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-20y-6y=28-15$$

Tāpiri -20y ki te -6y.

$$-26y=28-15$$

Tāpiri 28 ki te -15.

$$-26y=13$$

Whakawehea ngā taha e rua ki te -26.

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

Whakaurua te -\frac{1}{2} mō y ki 4x+2y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+2\left(-\frac{1}{2}\right)=5$$

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

$$4x-1=5$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$4x=6$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{3}{2}$$

Kua oti te pūnaha te whakatau.

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